Integration - Find the missing hypothesis: let f : [a,b] → R be a function. Then F : [a,b] → R given by F(x):=∫ax f is differentiable and F'=f., Sequences of functions - Is there a sequence (f_n : [0,1] → R) converging to f pointwise, where each f_n has a finite number of discontinuities but f is not integrable? What if the convergence is uniform?, Sequences of functions - Is there a sequence (f_n : [0,1] → R) converging to f uniformly, where each f_n is not integrable, but f is integrable., Continuity - Let f : R → R be a continuous function. Suppose that f(x) is irrational for all x∈R. What can you say about f?, Continuity / differentiability - Let f : R → R be a differentiable function. Suppose that f(x)>|x| for all x∈R. Show that there exists c∈R such that f'(c)=0., Differentiability / Integration - Let f : [0,1] → R be differentiable. Let f' : [0,1] → R be continuous. Show that there is M>0 such that |∫0xf(t)dt-f(0)x|≤Mx2 for all x∈[0,1].  , Metric spaces - Consider the uniform metric d∞ on the set of continuous functions C0[-1,1]. Let A⊂C0[-1,1] be the subset of differentiable functions. Is A closed? Is A open? , Integration - Let f : [0,1] → R be defined by f(x)=1/(1+x2). Let P={0,1/3,2/3,1}. Compute U(f,P) and L(f,P). Use this to show 272/390 < π/4 <337/390., Metric spaces - Consider the uniform metric on the set C0[-1,1]. Let A = {f∈C0[-1,1] : f(-1)=f(1)}. Is A a closed subset of C0[-1,1]??, Series of functions - Find the missing hypothesis: if for all k ∈ N, gk : [0,1] → R is differentiable, then f=∑∞k=1gk is differentiable and f'=∑∞k=1g'k. Do you remember a way to test the hypothesis?,

Recap Final Mathematical Analysis

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