Math 461 Bard College

# Homework 4

Due Date: Friday, February 26

1. Let $$f\colon\mathbb{R} \to\mathbb{R}$$. For each $$x\in\mathbb{R}$$, the oscillation of $$f$$ at $$x$$ is defined by $\mathrm{osc}_f(x) \,=\, \inf\bigl\{\mathrm{diam}\bigl(f(I)\bigr) \;\bigl|\; I\text{ is an open interval containing }x\bigr\},$ where $\mathrm{diam}(S) \,=\, \sup\bigl\{|x-y| \;\bigl|\; x,y\in S\bigr\}$ for any set $$S\subseteq\mathbb{R}$$.
1. If $$x\in\mathbb{R}$$, prove that $$f$$ is continuous at $$x$$ if and only if $$\mathrm{osc}_f(x) = 0$$.
2. Prove that the set $U_b \,=\, \{x\in \mathbb{R} \mid \mathrm{osc}_f(x) < b\}$ is open for every $$b\in(0,\infty)$$.
3. Use parts (a) and (b) to prove that $C \,=\, \{x\in \mathbb{R} \mid f\text{ is continuous at }x\}.$ is a $$G_\delta$$ set, and hence measurable.
2. Let $$\{f_n\}$$ be a sequence of continuous functions from $$\mathbb{R}$$ to $$\mathbb{R}$$.
1. Prove that the set $\{x \in \mathbb{R} \mid f_n(x) \to 0\text{ as }n\to\infty\}$ is a Borel set, and hence measurable.
2. Prove that the set $\bigl\{x \in \mathbb{R} \;\bigl|\; \text{the sequence }\{f_n(x)\}\text{ converges}\}$ is a Borel set, and hence measurable.