Continuous Real-Valued Functions of a Real Variable

Let $E$ be a closed set of real numbers and $f$ a real-valued function that is defined and continuous on $E$. Show that there is a function $g$ defined and continuous on all of $\mathbb {R}$ such that $f(x) = g(x)$ for each $x \in E$.

Define the real-valued function $f$ on $\mathbb {R}$ by setting

At what points is $f$ continuous?

Let $f$ and $g$ be continuous real-valued functions with a common domain $E$.

Show that a Lipschitz function is uniformly continuous but there are uniformly continuous functions that are not Lipschitz.

A continuous function $\varphi$ on $[a, b]$ is called piecewise linear provided there is a partition $a = x_{0} < x_{1} < \cdots < x_{n} = b$ of $[a, b]$ for which $\varphi$ is linear on each interval $\left[x_{i}, x_{i+1}\right]$. Let $f$ be a continuous function on $[a, b]$ and $\epsilon$ a positive number. Show that there is a piecewise linear function $\varphi$ on $[a, b]$ with $\left|f(x)-\varphi (x)\right| < \epsilon$ for all $x \in [a, b]$.

Show that a non-empty set $E$ of real numbers is closed and bounded if and only if every continuous real-valued function on $E$ takes a maximum value.

Show that a set $E$ of real numbers is closed and bounded if and only if every open cover of $E$ has a finite subcover.

Show that a non-empty set $E$ of real numbers is an interval if and only if every continuous real-valued function on $E$ has an interval as its image.

Show that a monotone function on an open interval is continuous if and only if its image is an interval.

Let $f$ be a real-valued function defined on $\mathbb {R}$. Show that the set of points at which $f$ is continuous is a $G_{\delta }$ set.

Let $\left\{ f_{n}\right\}$ be a sequence of continuous functions defined on $\mathbb {R}$. Show that the set of points $x$ at which the sequence $\left\{ f_{n}(x)\right\}$ converges to a real number is the intersection of a countable collection of $F_{\sigma }$ sets.

Let $f$ be a continuous real-valued function on $\mathbb {R}$. Show that the inverse image with respect to $f$ of an open set is open, of a closed set is closed, and of a Borel set is Borel.

A sequence $\left\{ f_{n}\right\}$ of real-valued functions defined on a set $E$ is said to converge uniformly on $E$ to a function $f$ if given $\epsilon > 0$, there is an $N$ such that for all $x \in E$ and all $n \geq N$, we have $\left|f_{n}(x)-f(x)\right| < \epsilon$. Let $\left\{ f_{n}\right\}$ be a sequence of continuous functions defined on a set $E$. Prove that if $\left\{ f_{n}\right\}$ converges uniformly to $f$ on $E$, then $f$ is continuous on $E$.