Real analysis#
Some notes on real analysis, following the textbook Understanding analysis
About this chapter
Chapter-by-chapter notes on the textbook Understanding Analysis. Everything starts from the axiom of completeness in chapter 1: it is what separates \(\mathbb{R}\) from \(\mathbb{Q}\), and chapter 2 shows it is equivalent to the nested interval property, monotone convergence, Bolzano-Weierstrass and the Cauchy criterion.
Later chapters reuse the same \(\epsilon\)-style definitions for limits of functions, continuity and derivatives. Chapter 6 asks when a limit of functions keeps their properties, and the answer is usually uniform convergence. The overview at the end lines up the parallel theorems (algebraic limit theorems, sequential criteria, limit theorems) across chapters.
Chapter outline: 8 sections
- Ch 1 - the real numbersThe rationals have gaps (no rational squares to 2).
- Ch 2 - sequences and series1 interactiveConvergence is defined with an \epsilon and an N: past some index, every term is within \epsilon of the limit.
- Ch 3 - basic topology of ROpen, closed and compact sets.
- Ch 4 - functional limits and continuityThe \epsilon-N definition of chapter 2 becomes an \epsilon-\delta definition for \lim_{x \to c} f(x).
- Ch 5 - the derivativeThe derivative is a functional limit of difference quotients.
- Ch 6 - sequences and series of function1 interactivePointwise convergence fixes x and sends n to infinity.
- Ch 7 - the Riemann IntegralThe integral is built from upper and lower sums over partitions.
- OverviewThe same few ideas repeated across chapters, listed side by side.
Ch 1 - the real numbers#
The rationals have gaps (no rational squares to 2). The axiom of completeness fills them: every set bounded above has a least upper bound. Countability arguments then show \(\mathbb{Q}\) is countable and \(\mathbb{R}\) is not.
there is no rational number whose square is 2 (proof by contradiction)
contrapositive: \(-q \to -p\) - logically equivalent
triangle inequality: \(\vert a+b\vert \leq \vert a\vert + \vert b\vert\) (often use |a-b| = |(a-c)+(c-b)|)
axiom of completeness - every nonempty set \(A \subseteq \mathbb{R}\) that is bounded above has a least upper bound
doesn’t work for \(\mathbb{Q}\)
supremum = supA = least upper bound (similarly, infimum)
supA is an upper bound of A
if \(s \in \mathbb{R}\) is another u.b. then \(s \geq supA\)
can be restated as \(\forall \epsilon > 0, \exists a \in A\) \(s-\epsilon < a\)
nested interval property - for each \(n \in N\), assume we are given a closed interval \(I_n = [a_n,b_n]=\{ x \in \mathbb{R} : a_n \leq x \leq b_n \}\) Assume also that each \(I_n\) contains \(I_{n+1}\). Then, the resulting nested sequence of nonempty closed intervals \(I_1 \supseteq I_2 \supseteq ...\) has a nonempty intersection use AoC with x = sup{\(a_n: n \in \mathbb{N}\)} in the intersection of all sets
archimedean property
\(\mathbb{N}\) is unbounded above (sup \(\mathbb{N}=\infty\))
\(\forall x \in \mathbb{R}, x>0, \exists n \in \mathbb{N}, 0 < \frac{1}{n} < x\)
\(\mathbb{Q}\) is dense in \(\mathbb{R}\) - for every \(a,b \in \mathbb{R}, a<b\), \(\exists r \in \mathbb{Q}\) s.t. \(a<r<b\)
pf: want \(a < \frac{m}{n} < b\)
by Archimedean property, want \(\frac{1}{n} < b-a\)
corollary: the irrationals are dense in \(\mathbb{R}\)
there exists a real number \(r \in \mathbb{R}\) satisfying \(r^2 = 2\)
pf: let r = \(\sup \{ t \in \mathbb{R} : t^2 < 2 \}\). disprove \(r^2<2, r^2>2\) by considering \(r+\frac{1}{n},r-\frac{1}{n}\)
A ~ B if there exists f:A->B that is 1-1 and onto
A is finite - there exists n \(\in \mathbb{N}\) s.t. \(\mathbb{N}_n\)~A
countable = \(\mathbb{N}\)~A.
uncountable - inifinite set that isn’t countable
Q is countable
pf: Let \(A_n = \{ \pm \frac{p}{q}:\) where p,q \(\in \mathbb{N}\) are in lowest terms with p+q=n}
R is uncountable
pf: Assume we can enumerate \(\mathbb{R}\) Use NIP to exclude one point from \(\mathbb{R}\) each time. The intersection is still nonempty, so we didn’t succesfully enumerate \(\mathbb{R}\)
\(\frac{x}{x^2-1}\) maps (0,1) \(\to \mathbb{R}\)
countable union of countable sets is countable
if \(A \subseteq B\) and B countable, then A is either countable or finite
if \(A_n\) is a countable set for each \(n \in \mathbb{N}\), then their union is countable
the open interval (0,1) = \(\{ x \in \mathbb{R} : 0 < x < 1 \}\) is uncountable
pf: diagonalization - assume there exists a function from (0,1) to \(\mathbb{R}\). List the decimal expansions of these as rows of a matrix. Complement of diagonal does not exist.
cantor’s thm - Given any set A, there does not exist a function f:\(A \to P(A)\) that is onto
P(A) is the set of all subsets of A
Ch 2 - sequences and series#
Convergence is defined with an \(\epsilon\) and an \(N\): past some index, every term is within \(\epsilon\) of the limit. A series converges when its sequence of partial sums does.
a sequence \((a_n)\) converges to a real number if \(\forall \epsilon > 0, \exists N \in \mathbb{N}\) such that \(\forall n\geq N, \vert a_n-a\vert < \epsilon\)
otherwise it diverges
if a limit exists, it is unique
a sequence \((x_n)\) is bounded if there exists a number M > 0 such that \(\vert x_n\vert \leq M \forall n \in \mathbb{N}\)
every convergent sequence is bounded
algebraic limit thm - let lim \(a_n = a\) and lim \(b_n\) = b. Then
lim(\(ca_n\)) = ca
lim(\(a_n+b_n\)) = a+b
lim(\(a_n b_n\)) = ab
lim(\(a_n/b_n\)) = a/b, provided b \(\neq\) 0
pf 3: use triangle inequality, \(\vert a_nb_n-ab\vert =\vert \cblue{a_nb_n-ab_n}+\corange{ab_n-ab}\vert =...=\cblue{\vert b_n\vert \vert a_n-a\vert} +\corange{\vert a\vert \vert b_n-b\vert}\)
pf 4: show \((b_n) \to b\) implies \((\frac{1}{b_n}) \to \frac{1}{b}\)
order limit thm - Assume lim \(a_n = a\) and lim \(b_n\) = b.
If \(a_n \geq 0\) \(\forall n \in \mathbb{N}\), then \(a \geq 0\)
If \(a_n \leq b_n\) \(\forall n \in \mathbb{N}\), then \(a \leq b\)
If \(\exists c \in \mathbb{R}\) for which \(c \leq b_n\) \(\forall n \in \mathbb{N}\), then \(c \leq b\)
pf 1: by contradiction
monotone - increasing or decreasing (not strictly)
monotone convergence thm - if a sequence is monotone and bounded, then it converges
convergence of a series
define \(s_m=a_1+a_2+...+a_m\)
\(\sum_{n=1}^\infty a_n\) converges to A \(\iff (s_m)\) converges to A
cauchy condensation test - suppose \(a_n\) is decreasing and satisfies \(a_n \geq 0\) for all \(n \in \mathbb{N}\). Then, the series \(\sum_{n=1}^\infty a_n\) converges iff the series \(\sum_{n=1}^\infty 2^na_{2^n}\) converges
p-series \(\sum_{n=1}^\infty 1/n^p\) converges iff p > 1
2.5#
Subsequences inherit the limit of the sequence, which gives a quick divergence test: two subsequences with different limits. Bolzano-Weierstrass says every bounded sequence has a convergent subsequence.
let \((a_n)\) be a sequence and \(n_1<n_2<...\) be an increasing sequence of natural numbers. Then \((a_{n_1},a_{n_2},...)\) is a subsequence of \((a_n)\)
subsequences of a convergent sequence converge to the same limit as the original sequence
can be used as divergence criterion
bolzano-weierstrass thm - every bounded sequence contains a convergent subsequence
pf: use NIP, keep splitting interval into two
2.6#
A Cauchy sequence has terms that get close to each other, with no limit named. The Cauchy criterion says this is the same as converging, and completes the chain of equivalent forms of completeness.
\((a_n)\) is a cauchy sequence if \(\forall \epsilon > 0, \exists N \in \mathbb{N}\) such that \(\forall m,n\geq N, \vert a_n-a_m\vert < \epsilon\)
cauchy criterion - a sequence converges \(\iff\) it is a cauchy sequence
cauchy sequences are bounded
overview: AoC \(\iff\) NIP \(\iff\) MCT \(\iff\) BW \(\iff\) CC
2.7#
Series results follow from the sequence results applied to the partial sums \(s_m\). If a series converges, its terms go to 0; the tests below (comparison, geometric, absolute, alternating) give sufficient conditions the other way.
algebraic limit thm - let \(\sum_{n=1}^\infty a_n\) = A, \(\sum_{n=1}^\infty b_n\) = B
\(\sum_{n=1}^\infty ca_n\) = cA
\(\sum_{n=1}^\infty a_n+b_n\) = A+B
cauchy criterion for series - series converges \(\iff\) \((s_m)\) is a cauchy sequence
if the series \(\sum_{n=1}^\infty a_n\) converges then lim \(a_n=0\)
comparison test
geometric series - \(\sum_{n=0}^\infty a r^n = \frac{a}{1-r}\)
\(s_m = a+ar+...+ar^{m-1} = \frac{a(1-r^m)}{1-r}\)
absolute convergence test
alternating series test
decreasing
lim \(a_n\) = 0
then, \(\sum_{n=1}^\infty (-1)^{n+1} a_n\) converges
rearrangements: there exists one-to-one correspondence
if a series converges absolutely, any rearrangement converges to same limit
Ch 3 - basic topology of R#
Open, closed and compact sets. Heine-Borel says that in \(\mathbb{R}\) compact means closed and bounded, which is what the extreme value theorem in chapter 4 relies on.
3.1 cantor set#
The Cantor set has small length but is uncountable, a first example that size by length and size by cardinality can disagree.
C has small length, but its cardinality is uncountable
discussion of dimensions, doubling sizes leads to 2^dimension sizes
Cantor set is about dimension .631
3.2 open/closed sets#
Open sets are defined with \(\epsilon\)-neighborhoods and closed sets with limit points. The two are complements, and limit points can be described by sequences, which ties this chapter back to chapter 2.
A set O \(\subseteq \mathbb{R}\) is open if for all points a \(\in\) O there exists an \(\epsilon\)-neighborhood \(V_{\epsilon}(a) \subseteq O\)
\(V_{\epsilon}(a)=\{ x \in R : \vert x-a\vert < \epsilon\)}
the union of an arbitrary collection of open sets is open
the intersection of a finite collection of open sets is open
a point x is a limit point of a set A if every \(\epsilon\)-neighborhood \(V_{\epsilon}(x)\) of x intersects the set A at some point other than x
a point x is a limit point of a set A if and only if x = lim \(a_n\) for some sequence (\(a_n\)) contained in A satisfying \(a_n \neq x\) for all n \(\in\) N
isolated point - not a limit point
set \(F \subseteq \mathbb{R}\) closed - contains all limit points
closed iff every Cauchy sequence contained in F has a limit that is also an element of F
density of Q in R - for every \(y \in \mathbb{R}\), there exists a sequence of rational numbers that converges to y
closure - set with its limit points
closure \(\bar{A}\) is smallest closed set containing A
iff set open, complement is closed
R and \(\emptyset\) are both open and closed
the union of a finite collection of closed sets is closed
the intersection of an arbitrary collection of closed sets is closed
3.3#
Compactness is defined through sequences: every sequence in \(K\) has a subsequence converging inside \(K\). Heine-Borel gives two equivalent forms, closed and bounded, and every open cover has a finite subcover.
a set K \(\subseteq \mathbb{R}\) is compact if every sequence in K has a subsequence that converges to a limit that is also in K
Nested Compact Set Property - intersection of nested sequence of nonempty compact sets is not empty
let A \(\subseteq \mathbb{R}\). open cover for A is a (possibly infinite) collection of open sets whose union contains the set A.
given an open cover for A, a finite subcover is a finite sub-collection of open sets from the original open cover whose union still manages to completely contain A
Heine-Borel thm - let K \(\subseteq \mathbb{R}\). All of the following are equivalent
K is compact
K is closed and bounded
every open cover for K has a finite subcover
Ch 4 - functional limits and continuity#
The \(\epsilon\)-\(N\) definition of chapter 2 becomes an \(\epsilon\)-\(\delta\) definition for \(\lim_{x \to c} f(x)\). The sequential criterion translates each functional limit back into limits of sequences.
4.1#
dirichlet function: 1 if r \(\in \mathbb{Q}\) 0 otherwise
4.2 functional limits#
Two equivalent definitions, one with \(\epsilon\) and \(\delta\) and one with neighborhoods. The sequential criterion is the usual tool for showing a limit does not exist.
def 1. Let f:\(A \to R\), and let c be a limit point of the domain A. We say that \(\lim_{x \to c} f(x) = L\) provided that for all \(\epsilon\) > 0, there exists a \(\delta\) > 0 s.t. whenever 0 < |x-c| < \(\delta\) (and x \(\in\) A) it follows that |f(x)-L|< \(\epsilon\)
def 2. Let f:\(A \to R\), and let c be a limit point of the domain A. We say that \(\lim_{x \to c} f(x) = L\) provided that for every \(\epsilon\)-neighborhood \(V_{\epsilon}(L)\) of L, there exists a \(\delta\)-neighborhood \(V_{\delta}(\)c) around c with the property that for all x \(\in V_{\delta}(\)c) different from c (with x \(\in\) A) it follows that f(x) \(\in V_{\epsilon}(L)\).
sequential criterion for functional limits - Given function f:\(A \to R\) and a limit point c of A, the following 2 statements are equivalent:
\(\lim_{x \to c} f(x) = L\)
for all sequences \((x_n) \subseteq\) A satisfying \(x_n \neq\) c and \((x_n) \to c\), it follows that \(f(x_n) \to L\).
algebraic limit thm for functional limits
divergence criterion for functional limits
4.3 continuous functions#
Continuity at \(c\) is the functional limit definition with \(L = f(c)\) and \(x = c\) allowed. Compositions of continuous functions are continuous.
a function f:\(A \to R\) is continuous at a point c \(\in\) A if, for all \(\epsilon\)>0, there exists a \(\delta\)>0 such that whenever |x-c|<\(\delta\) (and x\(\in\) A) it follows that \(\vert f(x)-f( c)\vert <\epsilon\). F is continous if it is continuous at every point in the domain A
characterizations of continuouty
criterion for discontinuity
algebraic continuity theorem
if f is continuous at c and g is continous at f( c) then g \(\circ\) f is continuous at c
4.4 continuous functions on compact sets#
Continuous functions send compact sets to compact sets, which gives the extreme value theorem. On a compact set, continuity also upgrades to uniform continuity, where one \(\delta\) works for every point.
preservation of compact sets - if f continuous and K compact, then f(K) is compact as well
extreme value theorem - if f if continuous on a compact set K, then f attains a maximum and minimum value. In other words, there exist \(x_0,x_1 \in K\) such that \(f(x_0) \leq f(x) \leq f(x_1)\) for all x \(\in\) K
f is uniformly continuous on A if for every \(\epsilon\)>0, there exists a \(\delta\)>0 such that for all x,y \(\in\) A, \(\vert x-y\vert < \delta \implies \vert f(x)-f(y)\vert < \epsilon\)
a function f fails to be uniformly continuous on A iff there exists a particular \(\epsilon_o\) > 0 and two sequences \((x_n),(y_n)\) in A sastisfying \(\vert x_n - y_n\vert \to 0\) but \(\vert f(x_n)-f(y_n)\vert \geq \epsilon_o\)
a function that is continuous on a compact set K is uniformly continuous on K
4.5 intermediate value theorem#
intermediate value theorem - Let f:[a,b]\( \to R\) be continuous. If L is a real number satisfying f(a) < L < f(b) or f(a) > L > f(b), then there exists a point c \(\in (a,b)\) where f( c) = L
a function f has the intermediate value property on an inverval [a,b] if for all x < y in [a,n] and all L between f(x) and f(y), it is always possible to find a point c \(\in (x,y)\) where f( c)=L.
Ch 5 - the derivative#
The derivative is a functional limit of difference quotients. Its main results are the mean value theorems, which turn information about \(f'\) into information about \(f\).
5.2 derivatives and the intermediate value property#
Differentiable implies continuous, and the usual rules (product, quotient, chain) follow from the algebraic limit theorem. Darboux’s theorem says a derivative has the intermediate value property even when it is not continuous.
let g: A -> R be a function defined on an interval A. Given c \(\in\) A, the derivative of g at c is defined by g’( c) = \(\lim_{x \to c} \frac{g(x) - g( c)}{x-c}\), provided this limit exists. Then g is differentiable at c. If g’ exists for all points in A, we say g is differentiable on A
identity: \(x^n-c^n = (x-c)(x^{n-1}+cx^{n-2}+c^2x^{n-3}+...+c^{n-1}\))
differentiable \(\implies\) continuous
algebraic differentiability theorem
adding
scalar multiplying
product rule
quotient rule
chain rule: let f:A-> R and g:B->R satisfy f(A)\(\subseteq\) B so that the composition g \(\circ\) f is defined. If f is differentiable at c in A and g differentiable at f( c) in B, then g \(\circ\) f is differnetiable at c with (g\(\circ\)f)’( c)=g’(f( c))*f’( c)
interior extremum thm - let f be differentiable on an open interval (a,b). If f attains a maximum or minimum value at some point c \(\in\) (a,b), then f’( c) = 0.
Darboux’s thm - if f is differentiable on an interval [a,b], and a satisfies f’(a) < \(\alpha\) < f’(b) (or f’(a) > \(\alpha\) > f’(b)), then there exists a point c \(\in (a,b)\) where f’( c) = \(\alpha\)
derivative satisfies intermediate value property
5.3 mean value theorems#
The mean value theorem finds a point where the slope equals the average slope over \([a, b]\). Its consequences: \(f' = 0\) forces \(f\) constant, and two functions with the same derivative differ by a constant. L’Hospital’s rules close the section.
mean value theorem - if f:[a,b] -> R is continuous on [a,b] and differentiable on (a,b), then there exists a point c \(\in\) (a,b) where \(f'( c) = \frac{f(b)-f(a)}{b-a}\)
Rolle’s thm - f(a)=f(b) -> f’( c)=0
if f’(x) = 0 for all x in A, then f(x) = k for some constant k
if f and g are differentiable functions on an interval A and satisfy f’(x) = g’(x) for all x \(\in\) A, then f(x) = g(x) + k for some constant k
generalized mean value theorem - if f and g are continuous on the closed interval [a,b] and differentiable on the open interval (a,b), then there exists a point c \(\in (a,b)\) where |f(b)-f(a)|g’( c) = |g(b)-g(a)|f’( c). If g’ is never 0 on (a,b), then can be restated \(\frac{f'( c)}{g'( c)} = \frac{f(b)-f(a)}{g(b)-g(a)}\)
given g: A -> R and a limit point c of A, we say that \(\lim_{x \to c} g(x) = \infty\) if, for every M > 0, there exists a \(\delta\)> 0 such that whenever 0 < |x-c| < \(\delta\) it follows that g(x) ≥ M
L’Hospital’s Rule: 0/0 - let f and g be continuous on an interval containing a, and assume f and g are differentiable on this interval with the possible exception of the point a. If f(a) = g(a) = 0 and g’(x) ≠ 0 for all x ≠ a, then \(\lim_{x \to a} \frac{f'(x)}{g'(x)} = L \implies \lim_{x \to a} \frac{f'(x)}{g'(x)} = L\)
L’Hospital’s Rule: \(\infty / \infty\) - assume f and g are differentiable on (a,b) and g’(x) ≠ 0 for all x in (a,b). If \(\lim_{x \to a} g(x) = \infty \), then \(\lim_{x \to a} \frac{f'(x)}{g'(x)} = L \implies \lim_{x \to a} \frac{f'(x)}{g'(x)} = L\)
Ch 6 - sequences and series of function#
Pointwise convergence fixes \(x\) and sends \(n\) to infinity. Uniform convergence asks for a single \(N\) that works for every \(x\) at once. Uniform convergence is what carries continuity and integrals over to the limit; for derivatives, the derivatives themselves must converge uniformly.
6.2 uniform convergence of a sequence of functions#
for each n \(\in \mathbb{N}\) let \(f_n\) be a function defined on a set A\(\subseteq R\). The sequence (\(f_n\)) of functions converges pointwise on A to a function f if, for all x in A, the sequence of real numbers \(f_n(x)\) converges to f(x)
let (\(f_n\)) be a sequence of functions defined on a set A\(\subseteq\)R. Then (\(f_n\)) converges unformly on A to a limit function f defined on A if, for every \(\epsilon\)>0, there exists an N in \(\mathbb{N}\) such that \(\forall n ≥N, x \in A , \vert f_n(x)-f(x)\vert <\epsilon\)
Cauchy Criterion for uniform convergence - a sequence of functions \((f_n)\) defined on a set A \(\subseteq\) R converges uniformly on A iff \(\forall \epsilon > 0 \exists N \in \mathbb{N}\) s.t. whenever m,n ≥N and x in A, \(\vert f_n(x)-f_m(x)\vert <\epsilon\)
continuous limit thm - Let (\(f_n\)) be a sequence of functions defined on A that converges uniformly on A to a function f. If each \(f_n\) is continuous at c in A, then f is continuous at c
6.3 uniform convergence and differentiation#
Uniform convergence of \(f_n\) is not enough to differentiate the limit. What is needed is uniform convergence of the derivatives \(f_n'\) plus convergence of \(f_n\) at a single point.
differentiable limit theorem - let \(f_n \to f\) pointwise on the closed interval [a,b], and assume that each \(f_n\) is differentiable. If \((f'_n)\) converges uniformly on [a,b] to a function g, then the function f is differentiable and f’=g
let (\(f_n\)) be a sequence of differentiable functions defined on the closed interval [a,b], and assume \((f'_n)\) converges uniformly to a function g on [a,b]. If there exists a point \(x_0 \in [a,b]\) for which \(f_n(x_0)\) is convergent, then (\(f_n\)) converges uniformly. Moreover, the limit function f = lim \(f_n\) is differentiable and satisfies f’ = g
6.4 series of functions#
The results of 6.2 and 6.3 applied to partial sums of \(\sum f_n\). The Weierstrass M-test is the practical way to check uniform convergence: bound each term by a constant \(M_n\) with \(\sum M_n\) finite.
term-by-term continuity thm - let \(f_n\) be continuous functions defined on a set A \(\subseteq\) R and assume \(\sum f_n\) converges uniformly on A to a function f. Then, f is continuous on A.
term-by-term differentiability thm - let \(f_n\) be differentiable functions defined on an interval A, and assume \(\sum f'_n(x)\) converges uniformly to a limit g(x) on A. If there exists a point \(x_0 \in [a,b]\) where \(\sum f_n(x_0)\) converges, then the series \(\sum f_n(x)\) converges uniformly to a differentiable function f(x) satisfying f’(x) = g(x) on A. In other words, \(f(x) = \sum f_n(x)\) and \(f'(x) = \sum f'_n(x)\)
Cauchy Criterion for uniform convergence of series - A series \(\sum f_n\) converges uniformly on A iff \(\forall \epsilon > 0 \exists N \in N\) s.t. whenever n>m≥N, x in A \(\vert f_{m+1}(x) + f_{m+2}(x) + f_{m+3}(x) + ...+f_n(x)\vert < \epsilon\)
Wierstrass M-Test - For each n in N, let \(f_n\) be a function defined on a set A \(\subseteq\) R, and let \(M_n > 0\) be a real number satisfying \(\vert f_n(x)\vert ≤ M_n\) for all x in A. If \(\sum M_n\) converges, then \(\sum f_n\) converges uniformly on A
6.5 power series#
A power series that converges at \(x_0\) converges absolutely inside \(\vert x \vert < \vert x_0 \vert\) and uniformly on compact sets there. The differentiated series converges on the same interval, so power series can be differentiated term by term as often as needed.
power series f(x) = \(\sum_{n=0}^\infty a_n x^n = a_0 + a_1 x_1 + a_2 x^2 + a_3 x^3 + ...\)
if a power series converges at some point \(x_0 \in \mathbb{R}\), then it converges absolutely for any x satisfying |x|<|\(x_0\)|
if a power series converges pointwise on the set A, then it converges uniformly on any compact set K \(\subseteq\) A
if a power series converges absolutely at a point \(x_0\), then it converges uniformly on the closed interval [-c,c], where c = |\(x_0\)|
Abel’s thm - if a power series converges at the point x = R > 0, the the series converges uniformly on the interval [0,R]. A similar result holds if the series converges at x = -R
if \(\sum_{n=0}^\infty a_n x^n\) converges for all x in (-R,R), then the differentiated series \(\sum_{n=0}^\infty n a_n x^{n-1}\) converges at each x in (-R,R) as well. Consequently the convergence is uniform on compact sets contained in (-R,R).
can take infinite derivatives
6.6 taylor series#
Taylor’s Formula \(\sum_{n=0}^\infty a_n x^n = a_0 + a_1 x_1 + a_2 x^2 + a_3 x^3 + ...\)
centered at 0: \(a_n = \frac{f^{(n)}(0)}{n!}\)
Lagrange’s Remainder thm - Let f be differentiable N+1 times on (-R,R), define \(a_n = \frac{f^{(n)}(0)}{n!}.....\)
not every infinitely differentiable function can be represented by its Taylor series (radius of convergence zero)
Ch 7 - the Riemann Integral#
The integral is built from upper and lower sums over partitions. A function is integrable when the two can be squeezed together, and the fundamental theorem links integrals back to derivatives.
7.2 def of Riemann integral#
Refining a partition raises lower sums and lowers upper sums. \(f\) is integrable when \(U(f) = L(f)\), or equivalently when some partition makes \(U(f,P) - L(f,P) < \epsilon\); every continuous function qualifies.
partition of [a,b] is a finite set of points from [a,b] that includes both a and b
lower sum - sum all the possible smallest rectangles
a partition Q is a refinement of a partition P if \(P \subseteq Q\)
if \(P \subseteq Q\), then L(f,P)≤L(f,Q) and U(f,P)≥U(f,Q)
a bounded function f on the interval [a,b] is Riemann-integrable if U(f) = L(f) = \(\int_a^b f\)
iff \(\forall \epsilon >0\), there exists a partition P of [a,b] such that \(U(f,P)-L(f,P)<\epsilon\)
U(f) = inf{U(f,P)} for all possible partitions P
if f is continuous on [a,b] then it is integrable
7.3 integrating functions with discontinuities#
if f:[a,b]->R is bounded and f is integrable on [c,b] for all c in (a,b), then f is integrable on [a,b]
7.4 properties of Integral#
assume f: [a,b]->R is bounded and let c in (a,b). Then, f is integrable on [a,b] iff f is integrable on [a,c] and [c,b]. In this case we have \(\int_a^b f = \int_a^c f + \int_c^b f.\)F
integrable limit thm - Assume that \(f_n \to f\) uniformly on [a,b] and that each \(f_n\) is integarble. Then, f is integrable and \(\lim_{n \to \infty} \int_a^b f_n = \int_a^b f\).
7.5 fundamental theorem of calculus#
Two halves: an antiderivative computes the integral, and the integral \(G(x)\) of a function continuous at \(c\) is differentiable at \(c\) with \(G'(c)\) equal to the integrand.
If f:[a,b] -> R is integrable, and F:[a,b]->R satisfies F’(x) = f(x) for all x \(\in\) [a,b], then \(\int_a^b f = F(b) - F(a)\)
Let f: [a,b]-> R be integrable and for x \(\in\) [a,b] define G(x) = \(\int_a^x g\). Then G is continuous on [a,b]. If g is continuous at some point \(c \in [a,b]\) then G is differentiable at c and G’(c) = g(c).
Overview#
The same few ideas repeated across chapters, listed side by side.
convergence
sequences
series
functional limits
normal, uniform
sequence of funcs
pointwise, uniform
series of funcs
pointwise, uniform
integrability
sequential criterion - usually good for proving discontinuous
limit points
functional limits
continuity
absence of uniform continuity
algebraic limit theorem ~ scalar multiplication, addition, multiplication, division
limit thm
sequences
series - can’t multiply / divide these
functional limits
continuity
differentiability
~integrability~
limit thms
continuous limit thm - Let (\(f_n\)) be a sequence of functions defined on A that converges uniformly on A to a function f. If each \(f_n\) is continuous at c in A, then f is continuous at c
differentiable limit theorem - let \(f_n \to f\) pointwise on the closed interval [a,b], and assume that each \(f_n\) is differentiable. If \((f'_n)\) converges uniformly on [a,b] to a function g, then the function f is differentiable and f’=g
convergent derivatives almost proves that \(f_n \to f\)
let (\(f_n\)) be a sequence of differentiable functions defined on the closed interval [a,b], and assume \((f'_n)\) converges uniformly to a function g on [a,b]. If there exists a point \(x_0 \in [a,b]\) for which \(f_n(x_0) \to f(x_0)\) is convergent, then (\(f_n\)) converges uniformly
integrable limit thm - Assume that \(f_n \to f\) uniformly on [a,b] and that each \(f_n\) is integarble. Then, f is integrable and \(\lim_{n \to \infty} \int_a^b f_n = \int_a^b f\).
functions are continuous at isolated points, but limits don’t exist there
uniform continuity: minimize \(\vert f(x)-f(y)\vert\)
derivative doesn’t have to be continuous
integrable if finite amount of discontinuities and bounded
Key ideas
The axiom of completeness separates \(\mathbb{R}\) from \(\mathbb{Q}\), and it is equivalent to the nested interval property, monotone convergence, Bolzano-Weierstrass and the Cauchy criterion.
Limits of sequences, functional limits, continuity and derivatives all use the same \(\epsilon\)-style definition, and sequential criteria translate each into statements about sequences.
In \(\mathbb{R}\), compact means closed and bounded, and continuous functions on compact sets attain their max and min and are uniformly continuous.
The mean value theorem turns facts about \(f'\) into facts about \(f\).
Pointwise limits can lose continuity; uniform convergence keeps continuity and integrals, and uniform convergence of the derivatives keeps differentiability.
A bounded function is Riemann-integrable when upper and lower sums can be squeezed together, which holds for every continuous function.