Basic Analysis I — Real Analysis · Problem 5.3.7

The Riemann Integral — Jiri Lebl

Mathematics — Real Analysis · fundamental-theorem-of-calculus

Problem

Let f:[a,b]→Rf : [a,b] \to \mathbb{R} be a continuous function. Let ϵ>0\epsilon > 0 be a constant so that a+ϵ<b−ϵa + \epsilon < b - \epsilon. For x∈[a+ϵ,b−ϵ]x \in [a+\epsilon, b-\epsilon], define g(x):=12ϵ∫x−ϵx+ϵf.g(x) := \frac{1}{2\epsilon}\int_{x-\epsilon}^{x+\epsilon} f. a) Show that gg is differentiable and find the derivative. b) Let ff be differentiable and fix x∈(a,b)x \in (a,b) (let ϵ\epsilon be small enough). What happens to g′(x)g'(x) as ϵ\epsilon gets smaller? c) Find gg for f(x):=∣x∣f(x) := |x|, ϵ=1\epsilon = 1 (you can assume [a,b][a,b] is large enough).

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