Analysis Final

Continuous subset of R

S subset of R is continuous if whenever U,V subset of R are open such that the intersection of S, U, V is empty and S is a subset of the union of U and V, then S is a subset of U or V

Continuity of a function at a point in R

For all n, there exists m such that |x-x₀|<1/m implies |f(x)-f(x₀)|<1/n

Continuity of a function on its domain in R

f is continuous on it’s domain if f is continuous on every point on the domain

Uniform continuity of a function on its domain in R

For all n, there exists m, for all x,y in D that |x-y|<1/m implies |f(x)-f(y)|<1/n

Characterization of continuity in terms of inverse images of open sets

A function on an open domain is continuous iff the inverse image of every open set is open

Limit of a function at a point in R

lim x→x₀ f(x)=y₀ iff for all n, there exists an m such that |x-x₀|<1/m implies |f(x)-y₀|<1/n

Intermediate Value Theorem in R

Let f:[a,b]→R be continuous. Then for any y between f(a) and f(b), there exists a c in [a,b] such that f(c)=y

Extreme Value Theorem in R

Let f:[a,b]→R be continuous. Then, there exists a c,d in [a,b] such that f(c)=max{f(x): x in [a,b]} and f(d)=min{f(x): x in [a,b]}

Uniform continuity theorem in R

A continuous function on a compact set is uniformly continuous

Differentiability of a function at a point in R

Let f:D→R where D contains a nhd of x₀. Then f is differentiable at x₀ with derivative f’(x₀) in R if for all n, there exists an m such that for all x in D, |x-x₀|<1/m implies |(f(x)-f(x₀))/(x-x₀)-f’(x₀)|<1/n

Big-O

g(x)=O|x-x0| as x→x0 if for all C>0, |x-x0|<1/m implies |g(x)|≤C|x-x₀|

little-o

g(x)=O|x-x0| as x→x0 if for all n there, exists an m such that |x-x0|<1/m implies |g(x)|≤1/n|x-x₀|

Continuously differentiable function on a domain in R

f is continuously differentiable on (a,b) if f is differentiable on (a,b) and f’ is continuous on (a,b)

Mean Value Theorem

If f is differentiable on (a,b) and continuous on [a.b] then f’(x)=(f(b)-f(a))/(b-a) for some x in (a,b)

Chain Rule in R

If f is differentiable at x₀ and g is differentiable at f(x₀) then gof is differentiable at x₀ and (gof)’(x₀)=g’(f(x₀))f’(x₀)

Taylor expansion of order n at a point of R

The Taylor expansion of order n at x₀ is for a Cⁿ function is defined by T_n(x₀,x)=Σn,k=0 1/(k!)f^k(x₀)(x-x₀)^k

Taylor’s theorem for Cⁿ function on R

If f is in Cⁿ, then f(x)-T_n(x₀,x)=o(|x-x₀|ⁿ)

Uniform convergence of a sequence of function on R

(f_j)_j=1 sequence of functions f_j:D→R, then f_j→f uniformly as j→∞ if for all n, there exists m such that for k>=m, for all x in D |f_k(x)-f(x)|<=1/n

Uniformly Cauchy sequence of functions on R

For all n, there exists m such that |f_j(x)-f_k(x)|<=1/n for all j,k>=m and all x in D

Cauchy criterion for uniform convergence

{f_n(x)} converges uniformly to some limit function iff for all n there exists m such that |fj(x)-fk(x)|<=1/n for all j,k>=m

Weierstrass approximation Theorem

Any continuous function on a compact interval is the uniform limit, on that interval, of polynomials

Weierstrass M-test

Let (M_j)SEQ([0,∞)). Suppose Σ∞,j=1(M_j)<∞. Suppose f_j:D→R satisfies sup|f_j(x)|<=M_j. Then Σ∞,j=1(f_j) converges uniformly in the sense that there exists F:D→R such that d(F, ΣN,j=1(f_j))→0 as N→∞

Rⁿ

The set of ordered n-tuples x=(x₁,…,x_n) of real numbers

Metric space

A metric space, M, is a set with real-value distance function d(x,y) for x,y in M satisfying:

1. d(x,y)>0 with equality iff x=y

2. d(x,y)=d(y,x)

3. d(x,z)<=d(x,y)+d(y,z)

Norm

A norm is a function ||x|| defined for every x in the vector space satisfying:

1. ||x||>0 with equality iff x=0

2. ||ax||=|a| ||x||

3. ||x+y||<=||x||+||y||

Open subset of a metric space

A subset A of a metric space M is open in M if every point of A lies in an open ball entirely contained in A

Limit point of a subset in a metric space

x is a limit point of a subset A if every nhd of x contains points of A not equal to x

Closed subset of a metric space

A subset A is closed if it contains all of its limit points

Cauchy sequence in metric space

{x_n} is cauchy if for all n there exists N suchthat d(x_j,x_k)<=1/n for all j,k>=N

Completeness property of a metric space

A metric space is complete if every Cauchy sequence has limit

Compact subset of a metric space

A is compact if every sequence of points in A has a limit point in A

Continuity of a function between metric spaces

Let M,N be metric spaces and f:M→N. For all n and x₀ in M there exists m such that d(x,x₀)<=1/m implies d(f(x),f(x₀))<=1/n

Uniform continuous function between metric spaces

f:M→N is uniformly continuous if for all n there exists m such that d(x,y)<=1/m implies d(f(x),f(y))<=1/n

Differentiability of a function Rⁿ and the notion of total derivative

Let f:D→R^m with D subset of Rⁿ open. f is differentiable if there exists on mxn matrix df(y) such that f(x)=f(y)+df(y)(x-y)+o(|x-y|) as x→y

Partial derivative

(∂f_k/∂x_y) exists at a point y if f_k(y+te_j)=f_k(y)+(∂f_k/∂x_y)yt+o(t) as t→0

Directional derivative

If u in Rⁿ, the directional derivative d_uf exists at y if f(y+tu)=f(y)+td_uf(y)+o(t) as t→0

Implicit function theorem (2 variables)

Suppose U subset R² is open, F:U→R is C’ and (∂F/∂y)(x₀,y₀)≠0. Then there exists a nhd U₁XU₂ of (x₀,y₀) and f:U₁→R in C’ such that {(x,y) in U₁XU₂: F(x,y)=F(x₀,y₀)}={(x,f(x): x in U}