Limits and Continuity

Let $f$ be a bounded, continuous, real-valued function on $\mathbb {R}^{2}$. Define the function $g$ on $\mathbb {R}$ by

Prove that $g$ is continuous.

Suppose that $f$ maps the compact interval $I$ into itself and that

for all $x, y \in I$, $x \neq y$. Can one conclude that there is some constant $M < 1$ such that, for all $x, y \in I$,

Let the real valued function $f$ on $[0,1]$ have the following two properties:

• If $[a,b] \subset [0,1]$, then $f([a,b])$ contains the interval with endpoints $f(a)$ and $f(b)$ (i.e., $f$ has the Intermediate Value Property).

• For each $c \in \mathbb {R}$, the set $f^{-1}(c)$ is closed.

Prove that $f$ is continuous.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ be uniformly continuous with $f(0) = 0$. Prove: there exists a positive number $B$ such that $\left|f(x)\right| \leq 1+B\left|x\right|$, for all $x$.

Suppose that $f$ is a continuous function on $\mathbb {R}$ which is periodic with period 1, i.e., $f(x+1) = f(x)$. Show:

Let $h: [0,1) \rightarrow \mathbb {R}$ be a function defined on the half-open interval $[0,1)$. Prove that if $h$ is uniformly continuous, there exists a unique continuous function $g: [0,1] \rightarrow \mathbb {R}$ such that $g(x) = h(x)$ for all $x \in [0,1)$.

Let $f$ be a continuous real valued function on $[0,\infty )$ such that $\lim_{x \rightarrow \infty } f(x)$ exists (finitely). Prove that $f$ is uniformly continuous.

Prove or supply a counterexample: If the function $f$ from $\mathbb {R}$ to $\mathbb {R}$ has both a left limit and a right limit at each point of $\mathbb {R}$, then the set of discontinuities of $f$ is, at most, countable.

Let $f: \mathbb {R} \rightarrow \mathbb {R}$ satisfy $f(x) \leq f(y)$ for $x \leq y$. Prove that the set where $f$ is not continuous is finite or countably infinite.

A function $f: [0,1] \rightarrow \mathbb {R}$ is said to be upper semicontinuous if given $x \in [0,1]$ and $\varepsilon > 0$, there exists a $\delta > 0$ such that if $\left|y-x\right| < \delta$, then $f(y) < f(x)+\varepsilon$. Prove that an upper semicontinuous function $f$ on $[0,1]$ is bounded above and attains its maximum value at some point $p \in [0,1]$.

Prove that a continuous function from $\mathbb {R}$ to $\mathbb {R}$ which maps open sets to open sets must be monotonic.

Let $f$ be a continuous function from $\mathbb {R}$ to $\mathbb {R}$ such that $\left|f(x)-f(y)\right| \geq \left|x-y\right|$ for all $x$ and $y$. Prove that the range of $f$ is all of $\mathbb {R}$.

Note: See also Problem 2.1.8.

Let $f$ be a continuous function on $[0,1]$. Evaluate the following limits.

Let $f$ be a function from $[0,1]$ into itself whose graph

is a closed subset of the unit square. Prove that $f$ is continuous.

Note: See also Problem 2.1.2.

Let $f$ be a continuous real valued function defined on $[0,1] \times [0,1]$. Let the function $g$ on $[0,1]$ be defined by

Prove that $g$ is continuous.

Let the function $f: \mathbb {R} \rightarrow \mathbb {R}$ be bounded on bounded sets and have the property that $f^{-1}(K)$ is closed whenever $K$ is compact. Prove $f$ is continuous.