3.3. real analysis

Some notes on real analysis, following the textbook Understanding analysis

3.3.1. ch 1 - the real numbers

  • there is no rational number whose square is 2 (proof by contradiction)

  • contrapositive: $\(-q \to -p\)$ - logically equivalent

  • triangle inequality: \(\|a+b\| \leq \|a\| + \|b\|\) (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)

    1. supA is an upper bound of A

    2. 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

    1. \(\mathbb{N}\) is unbounded above (sup \(\mathbb{N}=\infty\))

    2. \(\forall x \in \mathbb{R}, x>0, \exists n \in \mathbb{N}, 0 < \frac{1}{n} < x\)

  • \(\mathbb{Q}\) 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

3.3.2. ch 2 - sequences and series

  • a sequence \((a_n)\) converges to a real number if \(\forall \epsilon > 0, \exists N \in \mathbb{N}\) such that \(\forall n\geq N, \|a_n-a\| < \epsilon\)

    • 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 \(\|x_n\|\leq M \forall n \in \mathbb{N}\)

    • every convergent sequence is bounded

  • algebraic limit thm - let lim \(a_n = a\) and lim \(b_n\) = b. Then

    1. lim(\(ca_n\)) = ca

    2. lim(\(a_n+b_n\)) = a+b

    3. lim(\(a_n b_n\)) = ab

    4. lim(\(a_n/b_n\)) = a/b, provided b \(\neq\) 0

    • pf 3: use triangle inequality, \(\|a_nb_n-ab\|=\|a_nb_n-ab_n+ab_n-ab\|=...=\|b_n\|\|a_n-a\|+\|a\|\|b_n-b\|\)

    • 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.

    1. If \(a_n \geq 0\) \(\forall n \in \mathbb{N}\), then \(a \geq 0\)

    2. If \(a_n \leq b_n\) \(\forall n \in \mathbb{N}\), then \(a \leq b\)

    3. 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

3.3.2.1. 2.5

  • 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

3.3.2.2. 2.6

  • \((a_n)\) is a cauchy sequence if \(\forall \epsilon > 0, \exists N \in \mathbb{N}\) such that \(\forall m,n\geq N, \|a_n-a_m\| < \epsilon\)

  • 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

3.3.2.3. 2.7

  • algebraic limit thm - let \(\sum_{n=1}^\infty a_n\) = A, \(\sum_{n=1}^\infty b_n\) = B

    1. \(\sum_{n=1}^\infty ca_n\) = cA

    2. \(\sum_{n=1}^\infty a_n+b_n\) = A+B

  1. 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 \(a_n=0\)

  1. comparison test

  2. geometric series - \(\sum_{n=0}^\infty a r^n = \frac{a}{1-r}\)

    • \(s_m = a+ar+...+ar^{m-1} = \frac{a(1-r^m)}{1-r}\)

  3. absolute convergence test

  4. alternating series test

    1. decreasing

    2. 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

3.3.3. ch 3 - basic topology of R

3.3.3.1. 3.1 cantor set

  • 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

3.3.3.2. 3.2 open/closed sets

  • A set O \(\subseteq \mathbb{R}\) is open if for all points a \(\in\) O there exists an \(\epsilon\)-neighborhood \(V_{\epsilon}(a) \subseteq O\)

    • \(V_{\epsilon}(a)=\{ x \in R : \|x-a\| < \epsilon\)}

  1. the union of an arbitrary collection of open sets is open

  2. 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

  1. the union of a finite collection of closed sets is closed

  2. the intersection of an arbitrary collection of closed sets is closed

3.3.3.3. 3.3

  • 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

    1. K is compact

    2. K is closed and bounded

    3. every open cover for K has a finite subcover

3.3.4. ch 4 - functional limits and continuity

3.3.4.1. 4.1

  • dirichlet function: 1 if r \(\in \mathbb{Q}\) 0 otherwise

3.3.4.2. 4.2 functional limits

  • 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:

    1. \(lim_{x \to c} f(x) = L\)

    2. 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

3.3.4.3. 4.3 continuous functions

  • 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 \(\|f(x)-f( c)\|<\epsilon\). 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

3.3.4.4. 4.4 continuous functions on compact sets

  • 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, \(\|x-y\| < \delta \implies \|f(x)-f(y)\| < \epsilon\)

    • 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 \(\|x_n - y_n\| \to 0\) but \(\|f(x_n)-f(y_n)\| \geq \epsilon_o\)

  • a function that is continuous on a compact set K is uniformly continuous on K

3.3.4.5. 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.

3.3.5. ch 5 - the derivative

3.3.5.1. 5.2 derivatives and the intermediate value property

  • 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

    1. adding

    2. scalar multiplying

    3. product rule

    4. 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

3.3.5.2. 5.3 mean value theorems

  • 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 \(lim_{x \to a} \frac{f'(x)}{g'(x)} = L \implies lim_{x \to a} \frac{f'(x)}{g'(x)} = L\)

3.3.6. ch 6 - sequences and series of function

3.3.6.1. 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 \(\forall n ≥N, x \in A , \|f_n(x)-f(x)\|<\epsilon\)

    • 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, \(\|f_n(x)-f_m(x)\|<\epsilon\)

  • 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

3.3.6.2. 6.3 uniform convergence and differentiation

  • 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

3.3.6.3. 6.4 series of functions

  • 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 \(\|f_{m+1}(x) + f_{m+2}(x) + f_{m+3}(x) + ...+f_n(x)\| < \epsilon\)

    • 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 \(\|f_n(x)\| ≤ M_n\) for all x in A. If \(\sum M_n\) converges, then \(\sum f_n\) converges uniformly on A

3.3.6.4. 6.5 power series

  • 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

3.3.6.5. 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)

3.3.7. ch 7 - the Riemann Integral

3.3.7.1. 7.2 def of Riemann integral

  • 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

3.3.7.2. 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]

3.3.7.3. 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 \(lim_{n \to \infty} \int_a^b f_n = \int_a^b f\).

3.3.7.4. 7.5 fundamental theorem of calculus

  1. If f:[a,b] -> R is integrable, and F:[a,b]->R satisfies F’(x) = f(x) for all x \(\in\) [a,b], then \(\int_a^b f = F(b) - F(a)\)

  2. 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).

3.3.8. overview

  • convergence

    1. sequences

    2. series

    3. functional limits

      • normal, uniform

    4. sequence of funcs

      • pointwise, uniform

    5. series of funcs

      • pointwise, uniform

    6. integrability

  • sequential criterion - usually good for proving discontinuous

    1. limit points

    2. functional limits

    3. continuity

    4. absence of uniform continuity

  • algebraic limit theorem ~ scalar multiplication, addition, multiplication, division

    1. limit thm

    2. sequences

    3. series - can’t multiply / divide these

    4. functional limits

    5. continuity

    6. differentiability

    7. ~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 \(\|f(x)-f(y)\|\)

  • derivative doesn’t have to be continuous

  • integrable if finite amount of discontinuities and bounded