Lebl-Basic-Analysis-I · Problem 6.2.21

Sequences of Functions

Mathematics · uniform-convergence-derivative-discontinuous

Problem

Let fn(x):=x1+(nx)2f_n(x) \mathrel{:=} \dfrac{x}{1+(nx)^2}. Notice that fnf_n are differentiable functions. a) Show that {fn}n=1∞\{f_n\}_{n=1}^\infty converges uniformly to 00. b) Show that ∣fn′(x)∣≤1|f_n'(x)| \leq 1 for all xx and all nn. c) Show that {fn′}n=1∞\{f_n'\}_{n=1}^\infty converges pointwise to a function discontinuous at the origin. d) Let {an}n=1∞\{a_n\}_{n=1}^\infty be an enumeration of the rational numbers. Define gn(x):=∑k=1n2−kfn(x−ak).g_n(x) \mathrel{:=} \sum_{k=1}^n 2^{-k} f_n(x - a_k). Show that {gn}n=1∞\{g_n\}_{n=1}^\infty converges uniformly to 00. e) Show that {gn′}n=1∞\{g_n'\}_{n=1}^\infty converges pointwise to a function ψ\psi that is discontinuous at every rational number and continuous at every irrational number. In particular, lim⁡n→∞gn′(x)≠0\lim_{n\to\infty} g_n'(x) \neq 0 for every rational number xx.

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