Lebl-Basic-Analysis-I · Problem 3.2.18

Continuous Functions

Mathematics · continuous-functions

Problem

Suppose f:[−1,1]→Rf : [-1, 1] \to \mathbb{R} is a function continuous at all x∈[−1,1]∖{0}x \in [-1, 1] \setminus \{0\}. Show that for every ϵ\epsilon such that 0<ϵ<10 < \epsilon < 1, there exists a function g:[−1,1]→Rg : [-1, 1] \to \mathbb{R} continuous on all of [−1,1][-1, 1], such that f(x)=g(x)f(x) = g(x) for all x∈[−1,−ϵ]∪[ϵ,1]x \in [-1, -\epsilon] \cup [\epsilon, 1], and ∣g(x)∣≤∣f(x)∣|g(x)| \leq |f(x)| for all x∈[−1,1]x \in [-1, 1].

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