Basic Analysis I — Real Analysis · Problem 6.2.21

Sequences of Functions — Jiri Lebl

Mathematics — Real Analysis · uniform-convergence-derivatives-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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