1.
Prove that every constant function and the identity function \(x \mapsto x\) are continuous on \(\R\text{.}\)
Exercises 4.6 Exercises
2.
Prove that every function \(f \colon \Z \to \R\) is continuous.
3.
Let
\begin{equation*}
S=\{0\}\cup\left\{\frac1n : n \in \N\right\}
\cup \left\{2-\frac1n : n \in \N\right\}.
\end{equation*}
Determine all cluster points of \(S\) and all isolated points of \(S\text{.}\)
4.
Let \(S \subseteq \R\) and \(c \in \R\text{.}\) Prove that \(c\) is a cluster point of \(S\) if and only if there exists a sequence \((x_n)\) in \(S \setminus \{c\}\) such that \(x_n \to c\text{.}\)
5.
Prove directly from the \(\varepsilon\)-\(\delta\) definition that \(f(x)=x^2\) is continuous on \(\R\text{.}\)
Hint: for fixed \(c \in \R\text{,}\) write \(|x^2-c^2|=|x-c||x+c|\) and choose \(\delta \lt \min\{1,\varepsilon/(2|c|+1)\}\text{.}\)
6.
Prove directly from the definitions that
\begin{equation*}
\lim_{x\to 0} \frac{1}{x^2}=+\infty
\quad\text{and}\quad
\lim_{x\to +\infty} \frac{1}{x}=0.
\end{equation*}
7.
Define
\begin{equation*}
f(x)=\begin{cases}
-1 & \text{if } x \lt 0,\\
0 & \text{if } x=0,\\
1 & \text{if } x \gt 0.
\end{cases}
\end{equation*}
Compute \(\lim_{x\to 0^-} f(x)\) and \(\lim_{x\to 0^+} f(x)\text{.}\) Does \(\lim_{x\to 0} f(x)\) exist? Is \(f\) continuous at \(0\text{?}\)
8.
Define
\begin{equation*}
f(x)=\begin{cases}
\dfrac{x^2-4}{x-2} & \text{if } x \neq 2,\\
0 & \text{if } x=2.
\end{cases}
\end{equation*}
Find \(\lim_{x\to 2} f(x)\text{.}\) Is \(f\) continuous at \(2\text{?}\) How should \(f(2)\) be redefined to make \(f\) continuous at \(2\text{?}\)
9.
Let \(S \subseteq \R\) and let \(f \colon S \to \R\) be continuous. Prove that \(|f| \colon S \to \R\) is continuous.
10.
Let \(S \subseteq \R\text{,}\) and let \(f,g \colon S \to \R\) be continuous. Suppose that \(f(q)=g(q)\) for every \(q \in \Q \cap S\text{.}\) Prove that \(f=g\) on \(S\text{.}\)
11.
Let \(S \subseteq \R\text{,}\) let \(f \colon S \to \R\text{,}\) and let \(c \in S\text{.}\) Assume that \(f\) is continuous at \(c\) and \(f(c) \gt 0\text{.}\) Prove that there exists \(\delta \gt 0\) such that
\begin{equation*}
f(x) \gt \frac{f(c)}{2}
\end{equation*}
for every \(x \in S\) with \(|x-c| \lt \delta\text{.}\)
12.
Give examples showing that each hypothesis of the Extreme Value Theorem is necessary.
-
A continuous function on a bounded interval that does not attain an absolute maximum or an absolute minimum.
-
A continuous function on a closed interval that is not bounded above.
-
A function on a closed bounded interval that is not continuous and does not attain an absolute maximum or an absolute minimum.
13.
Use the Intermediate Value Theorem to prove that the equation \(\cos x = x\) has a solution in \((0,1)\text{.}\)
14.
Let \(I \subseteq \R\) be an interval, and let \(f \colon I \to \R\) be continuous. Prove that \(f(I)\) is an interval.
15.
Prove that every odd degree polynomial with real coefficients has a real root.
Hint: compare the dominant term for large positive and negative values of \(x\text{,}\) then apply the Intermediate Value Theorem.
16.
Prove that \(f(x)=x^2\) is not uniformly continuous on \(\R\text{.}\)
Hint: compare the values of \(f\) at \(n\) and \(n+\frac1n\text{.}\)
17.
Show that \(f(x)=\sin(1/x)\) is not uniformly continuous on \((0,1)\text{.}\)
Hint: use Proposition 4.5.5.
