1.
Let \(f: \R \to \R\) be defined by \(f(x) = |x|\text{,}\) with the Euclidean metric on both the domain and the codomain. Is \(f\) continuous at \(x=0\text{?}\) Prove your answer.
Exercises Exercises
2.
Let \(f: \R \to \R\) be defined by \(f(x) = \begin{cases}\frac{x}{|x|} \amp \text{ if } x \neq 0 \\ 1 \amp \text{ if } x=0. \end{cases}\) Is \(f\) continuous at \(x=0\text{?}\) Prove your answer.
3.
Let \((Y, d_Y) = (\R, d_E)\text{,}\) where \(d_E\) is the Euclidean metric.
(a)
Let \((X,d_X) = (\R^2, d_E)\text{.}\) Prove or disprove: the function \(f:X \to Y\) defined by \(f((x_1,x_2)) = x_1 + x_2\) is continuous.
(b)
Let \((X,d_X) = (\R^2, d_M)\) where \(d_M\) is the max metric. Prove or disprove: the function \(f:X \to Y\) defined by \(f((x_1,x_2)) = x_1 + x_2\) is continuous.
4.
Let \(X\) be any set and define \(d : X \times X \to \R\) by
\begin{equation*}
d(x,y) = \begin{cases}0 \amp \text{ if } x=y \\ 1 \amp \text{ if } x \neq y. \end{cases}
\end{equation*}
Exercise 1 asks us to show that \(d\) is a metric (the discrete metric) on \(\R\text{.}\) Let \((X,d_X)\) and \((Y, d_Y)\) be metric spaces with \(d_X\) the discrete metric. Determine all of the continuous functions \(f\) from \(X\) to \(Y\text{.}\)
5.
Let \(f\) and \(g\) be continuous functions from \((\R,d_E)\) to \((\R, d_E)\text{.}\)
(a)
Let \(k \in \R\) with \(k \neq 0\) and define \(kf : \R \to \R\) by \((kf)(x) = kf(x)\) for all \(x \in \R\text{.}\) Show that \(kf\) is a continuous function.
(b)
Define \(f+g : \R \to \R\) by \((f+g)(x) = f(x) + g(x)\) for all \(x \in \R\text{.}\) Show that \(f+g\) is a continuous function.
6.
Let \(f\) and \(g\) be continuous functions from \((\R, d_E)\) to \((\R,d_E)\text{.}\) In this exercise we will prove that \(fg\) is a continuous function from \(\R\) to \(\R\text{.}\) Let \(a\) be in \(\R\text{,}\) and follow the steps below to show that \(fg\) is continuous at \(x=a\text{.}\) Let \(\epsilon\) be a positive number.
(a)
We will first want to express \(f(x)g(x) - f(a)g(a)\) in a more useful way. Use the fact that \(f(x) = f(a) + (f(x)-f(a))\) and \(g(x) = g(a) + (g(x)-g(a))\) to show that
\begin{align*}
f(x)g(x)-f(a)g(a) = f(a)(g(x)-g(a)) \amp + g(a)(f(x)-f(a))\\
\amp + (f(x)-f(a))(g(x)-g(a))\text{.}
\end{align*}
(b)
Explain why there exist positive numbers \(\delta_1\text{,}\) \(\delta_2\text{,}\) \(\delta_3\text{,}\) and \(\delta_4\) such that
\begin{align*}
|f(x)-f(a)| \lt \sqrt{\frac{\epsilon}{3}} \amp \text{ when } |x-a| \lt \delta_1\\
|g(x)-g(a)| \lt \sqrt{\frac{\epsilon}{3}} \amp \text{ when } |x-a| \lt \delta_2\\
|f(x)-f(a)| \lt \frac{\epsilon}{3(1+|g(a)|)} \amp \text{ when } |x-a| \lt \delta_3\\
|g(x)-g(a)| \lt \frac{\epsilon}{3(1+|f(a)|)} \amp \text{ when } |x-a| \lt \delta_4\text{.}
\end{align*}
(c)
Use the results of (a) and (b) to show that \(fg\) is continuous at \(x=a\text{.}\) (Hint: \(1+|f(a)| \gt |f(a)|\text{.}\))
7.
Let \(f\) and \(g\) be functions from \((\R,d_E)\) to \((\R,d_E)\text{.}\)
(a)
Is it true that if \(f+g\) is a continuous function, then \(f\) and \(g\) are continuous functions? Verify your answer.
(b)
Is it true that if \(fg\) is a continuous function, then \(f\) and \(g\) are continuous functions? Verify your answer.
8.
Let \(f(x) = 2x^2+1\) map from \(\R\) to \(\R\text{,}\) with both the domain and codomain having the Euclidean metric.
(a)
Let \(\epsilon = \frac{1}{4}\text{.}\) Find a value of \(\delta\) such that \(|x-1| \lt \delta\) implies that \(|f(x)-f(a)| \lt \epsilon\text{.}\) You might use the applet at to confirm your value of \(\delta\text{.}\)
(b)
Prove that \(f\) is continuous at \(x=1\text{.}\)
9.
Define \(d: \R \times \R \to \R\) as
\begin{equation*}
d(x,y) = \min\{|x-y|,1\}\text{.}
\end{equation*}
Prove that \(d\) is a metric.
10.
Let \(f : \R \to \R\) be a continuous function, with both copies of \(\R\) having the Euclidean metric. Assume that \(f(x) = 0\) whenever \(x\) is rational. Prove that \(f(x) = 0\) for every \(x \in \R\text{.}\)
Hint.
Use Exercise 7.
11.
Let \(f: \R \to \R\) be defined by \(f(x) = 0\) if \(x\) is irrational and \(f(x) = 1\) if \(x\) is rational. Assume the Euclidean metric on both copies of \(\R\text{.}\) Show that \(f\) is not continuous at any point in \(\R\text{.}\)
Hint.
Use Exercise 7 and Exercise 10.
12.
Let \(g: \R \to \R\) be defined by \(g(x) = 0\) if \(x\) is irrational and \(g(x) = x\) if \(x\) is rational. Assume the Euclidean metric on both copies of \(\R\text{.}\) Show that \(g\) is continuous only at \(0\text{.}\)
13.
Let \(X\) be the set of continuous functions \(f: [a,b] \to \mathbb{R}\text{.}\) Let \(d^*\) be the distance function on \(X\) defined by
\begin{equation*}
d^*(f,g)= \int_a^b \la f(t)-g(t) \ra \, dt\text{,}
\end{equation*}
for \(f, g \in X\text{.}\) For each \(f \in X\text{,}\) set
\begin{equation*}
I(f) = \int_a^b f(t) \, dt\text{.}
\end{equation*}
(a)
Determine the value of \(d^*(f,g)\) when \(f(x) = x^2\text{,}\) \(g(x) = 3-2x\text{,}\) and \([a,b] = [-3,3]\text{.}\)
(b)
Determine the value of \(I(f)\) if \(f(x) = 2x\) and \([a,b] = [0,2]\text{.}\)
(c)
Prove that the function \(I : (X, d^*) \to (\R,d)\) is continuous, where \(d\) is the Euclidean metric.
Hint.
It helps to start by explicitly writing down what it means for \(I\) to be continuous in terms of the metrics \(d^*\) and \(d\) before trying to prove this statement.
14.
For each of the following, answer true if the statement is always true. If the statement is only sometimes true or never true, answer false and provide a concrete example to illustrate that the statement is false. If a statement is true, explain why.
(a)
Let \(f : X \to Y\) be a function, where \((X, d_X)\) and \((Y, d_Y)\) are metric spaces. If \(d_X\) is the discrete metric and \(d_Y\) is any metric, then \(f\) is continuous.
(b)
Let \(f : X \to Y\) be a function, where \((X, d_X)\) and \((Y, d_Y)\) are metric spaces. If \(d_Y\) is the discrete metric and \(d_X\) is any metric, then \(f\) is continuous.
(c)
Let \(d_1\) and \(d_2\) be two metrics on a set \(X\text{.}\) The identity function \(i_X : (X,d_1) \to (X,d_2)\) defined by \(i_X(x) = x\) for every \(x \in X\) is continuous.
(d)
Let \(f\) and \(g\) be continuous functions from \((\R^2, d_T)\) (the taxicab metric) to \((\R,d_E)\text{.}\) Then the function \(f+g\) from \((\R^2, d_T)\) to \((\R,d_E)\) defined by \((f+g)(x) = f(x) + g(x)\) for every \(x \in \R^2\) is a continuous function.
(e)
If \((X, d_X)\) and \((Y,d_Y)\) are metric spaces with \(y \in Y\text{,}\) then the constant function \(f: X \to Y\) defined by \(f(x) = y\) for every \(x \in X\) is a continuous function.
