# 3.3. real analysisÂ¶

Some notes on real analysis, following the textbook Understanding analysis

## 3.3.1. ch 1 - the real numbersÂ¶

• 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

• doesnâ€™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 isnâ€™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 didnâ€™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.

• cantorâ€™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 seriesÂ¶

• 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 $$c \leq b$$

• 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. 2.5Â¶

• 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. 2.6Â¶

• $$(a_n)$$ 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. 2.7Â¶

• 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 RÂ¶

### 3.3.3.1. 3.1 cantor setÂ¶

• 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 setsÂ¶

• 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. 3.3Â¶

• 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 continuityÂ¶

### 3.3.4.1. 4.1Â¶

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

### 3.3.4.2. 4.2 functional limitsÂ¶

• 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 functionsÂ¶

• 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 setsÂ¶

• 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 theoremÂ¶

• 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 derivativeÂ¶

### 3.3.5.1. 5.2 derivatives and the intermediate value propertyÂ¶

• 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 gâ€™( c) = $$\lim_{x \to c} \frac{g(x) - g( c)}{x-c}$$, provided this limit exists. Then g is differentiable at c. If gâ€™ 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

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)â€™( c)=gâ€™(f( c))*fâ€™( 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 fâ€™( c) = 0.

• Darbouxâ€™s thm - if f is differentiable on an interval [a,b], and a satisfies fâ€™(a) < $$\alpha$$ < fâ€™(b) (or fâ€™(a) > $$\alpha$$ > fâ€™(b)), then there exists a point c $$\in (a,b)$$ where fâ€™( c) = $$\alpha$$

• derivative satisfies intermediate value property

### 3.3.5.2. 5.3 mean value theoremsÂ¶

• 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 $$f'( c) = \frac{f(b)-f(a)}{b-a}$$

• Rolleâ€™s thm - f(a)=f(b) -> fâ€™( c)=0

• if fâ€™(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 fâ€™(x) = gâ€™(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)|gâ€™( c) = |g(b)-g(a)|fâ€™( c). If gâ€™ 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) â‰¥ M

• Lâ€™Hospitalâ€™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 gâ€™(x) â‰  0 for all x â‰  a, then $$lim_{x \to a} \frac{f'(x)}{g'(x)} = L \implies lim_{x \to a} \frac{f'(x)}{g'(x)} = L$$

• Lâ€™Hospitalâ€™s Rule: $$\infty / \infty$$ - assume f and g are differentiable on (a,b) and gâ€™(x) â‰  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 functionÂ¶

### 3.3.6.1. 6.2 uniform convergence of a sequence of functionsÂ¶

• 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 $$\forall n â‰¥N, x \in A , \|f_n(x)-f(x)\|<\epsilon$$

• 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 â‰¥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 differentiationÂ¶

• 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 fâ€™=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 fâ€™ = g

### 3.3.6.3. 6.4 series of functionsÂ¶

• 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 fâ€™(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>mâ‰¥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 $$\|f_n(x)\| â‰¤ M_n$$ 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 seriesÂ¶

• 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 = |$$x_0$$|

• Abelâ€™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 seriesÂ¶

• Taylorâ€™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: $$a_n = \frac{f^{(n)}(0)}{n!}$$

• Lagrangeâ€™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 IntegralÂ¶

### 3.3.7.1. 7.2 def of Riemann integralÂ¶

• 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)â‰¤L(f,Q) and U(f,P)â‰¥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 discontinuitiesÂ¶

• 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 IntegralÂ¶

• 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 calculusÂ¶

1. If f:[a,b] -> R is integrable, and F:[a,b]->R satisfies Fâ€™(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 Gâ€™(c) = g(c).

## 3.3.8. overviewÂ¶

• 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 - canâ€™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 fâ€™=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 donâ€™t exist there

• uniform continuity: minimize $$\|f(x)-f(y)\|$$

• derivative doesnâ€™t have to be continuous

• integrable if finite amount of discontinuities and bounded