MATH 4630‑01
Homework Assignment #5
DUE: Wednesday, Apr 17, 11:59p
Sections 4.3–4.5
Instructions: Show all work clearly and justify each conclusion.
Collaboration is encouraged, but write up your solutions individually in your own words.
For any prove/disprove problem: either give a proof, or give a specific counterexample (with a brief explanation of why it works).
For any prove/disprove problem: either give a proof, or give a specific counterexample (with a brief explanation of why it works).
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Provide an example of each or explain why it is impossible.
- Two functions $f$ and $g$, neither of which is continuous at $0$, but such that $f(x)g(x)$ and $f(x)+g(x)$ are continuous at $0$.
- A function $f(x)$ continuous at $0$ and a function $g(x)$ not continuous at $0$ such that $f(x)+g(x)$ is continuous at $0$.
- A function $f(x)$ continuous at $0$ and a function $g(x)$ not continuous at $0$ such that $f(x)g(x)$ is continuous at $0$.
- A function $f(x)$ not continuous at $0$ such that $[f(x)]^{2}$ is continuous at $0$.
- Assume $g:\mathbb{R}\to\mathbb{R}$ is continuous on $\mathbb{R}$ and let \[ K = \{\, x\in\mathbb{R} : g(x)=0 \,\}. \] Show that $K$ is a closed set.
- Let $f:\mathbb{R}\to\mathbb{R}$ be continuous at $c$ and suppose $f(c)>0$. Show that there exists a neighborhood $V_{\delta}(c)$ of $c$ such that if $x \in V_{\delta}(c)$, then $f(x)>0$. (Hint: Use the $\varepsilon$–$\delta$ definition of continuity of $f$ at $c$ with $\varepsilon = \tfrac{f(c)}{2}$.)
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Consider the function $f:\mathbb{R}\setminus\{0\}\to\mathbb{R}$ defined by
\[
f(x)=\frac{1}{x^{2}}.
\]
- Show that $f$ is uniformly continuous on $[1,\infty)$.
- Show that $f$ is not uniformly continuous on $(0,1]$.
- A function $f:A\to\mathbb{R}$ is said to be a Lipschitz function if there exists a constant $M>0$ with \[ \left|\frac{f(x)-f(y)}{x-y}\right|\le M \quad\text{for all } x\ne y. \] Show that if $f:A\to\mathbb{R}$ is Lipschitz, then $f$ is uniformly continuous on $A$.
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Provide an example of each of the following, or explain why it is impossible.
- A continuous function defined on an open interval with range equal to a closed interval.
- A continuous function defined on a closed interval with range equal to an open interval.
- A continuous function defined on an open interval with range equal to an unbounded closed set different from $\mathbb{R}$.
- A continuous function defined on all of $\mathbb{R}$ with range equal to $\mathbb{Q}$.
- Let $f$ be continuous on the interval $[0,1]$ to $\mathbb{R}$ and suppose that $f(0)=f(1)$. Prove that there exists a point $c\in[0,\tfrac{1}{2}]$ such that \[ f(c)=f\!\left(c+\tfrac{1}{2}\right). \]
This result shows that, at any given time, there exist antipodal points on the
Earth’s equator that have the same temperature.
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