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Fundamental Theorem of Calculus and Uniform Convergence

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Suppose f is Reimann integrable on [a,b] and let F(x)= ∫_a^x▒f(t)dt for all x ∈[a,b]. Prove that F is continuous on [a,b] (hint: f must be bounded)
Let F(x) = {█(x^2 sin⁡(1/x) if 0<|x|≤1,@0 if x=0)┤ and let f(x) = F'(x)
Prove that F'(x) exists for all x ∈[-1,1]
Find f(x) for all x ∈[-1,1], and prove that f is reimann integrable on [-1, 1]
Find ∫_(-1)^1▒f(x)dx

Let f_n(x) = 1/n sin(n^2 x)
Prove f_n converges uniformly on R to a differentiable function, yet 〖f'〗_n(0) diverges.

Let f_n (x)= 〖sin〗^n (x) for all x ∈[0,π]. Prove that 〖f'〗_n is not uniformly convergent on [0,π]. (hint: suppose false and deduce a contradiction)

Note: theorem included below to be used as an aid with the hint in #4

Suppose f_n is defined on a finite interval I and 〖f'〗_n is continuous on I. Suppose 〖f'〗_n converges uniformly on I. Suppose moreover that there exists at least one point a ∈ I such that f_n (a) is a convergent sequence of real numbers. Then there exists a differentiable function f such that f_n→f uniformly on I, and f'(x) ≡ lim┬(n→∞)⁡〖〖f^'〗_n (x)〗 on I

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Solution Preview

Suppose f is Reimann integrable on [a,b] and let F(x)= ∫_a^x▒f(t)dt for all x ∈[a,b]. Prove that F is continuous on [a,b] (hint: f must be bounded)
Consider F(x+h)= ∫_a^(x+h)▒f(t)dt= ∫_a^x▒f(t)dt+ ∫_x^(x+h)▒f(t)dt=F(x)+∫_x^(x+h)▒f(t)dt
Given that f is Reimann Integrable in [a, b] and hence is bounded on [a, b].
So, there exists some real number M such that for all x in [a, b] so in [x, x+h] for any h such that the second interval lies totally inside the first interval.
Taking the limit as ,
We have ...

Solution Summary

The fundamental theorem of calculus is applied.

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