Chapter 2

Show that \(F(x, y)=x^{2}+3 y\) is not uniformly continuous on the whole plane.

\((a)\) Find an interval on which \(f(x)=\frac{3 x}{x+4}\) is cont?nuous and 1 -to-1. (b) Find a formula for the corresponding inverse.

Show that if \(f\) is continuous on \(D\), then the set of points \(p\) where \(f(p) \leq C\) is closed relative to \(D\).

Discuss the continuity of the function \(f\) described by: (a) \(f(x)=\left\\{\begin{array}{ll}x \sin (1 / x) & x \neq 0 \\ 0 & x=0\end{array}\right.\) (b) \(f(x, y)=\frac{x y}{|x|+|y|} \quad\) for \((x, y) \neq(0,0)\) (c) \(f(x, y)=\frac{x^{2} y^{3}}{x^{4}+y^{6}} \quad\) for \((x, y) \neq(0,0)\) (d) \(f(x, y)=\left\\{\begin{array}{ll}\frac{x^{2}-y^{2}}{x-y} & \text { for } x \neq y \\ x-y & \text { when } x=y\end{array}\right.\)

Prove that \(Q(x, y)=x / y\) is continuous everywhere except on the line \(y=0\).

Find \(\lim _{x \rightarrow 2} \sqrt[3]{x}-\frac{3 / 2}{x-\sqrt{2}}\)

Prove that the function \(f(x)=1 /\left(1+x^{2}\right)\) is uniformly continuous on the whole line.

Let \(S=\left\\{\right.\) all \((x, y)\) with \(\left.3-\frac{x}{y}-\frac{y}{x} \leq 1\right\\}\). (a) Is \(S\) a closed set in the plane? (b) Does this conflict with Exercise \(1 ?\)

Let \(f(x)=\left\\{\begin{array}{l}1 \text { if } x \text { is a rational number } \\ 0 \text { if } x \text { is an irrational number. }\end{array}\right.\) Is \(f\) continuous anywhere?

Check that the function defined in \((2-2)\) is such that it is convergence preserving for all sequences of the form \(p_{n}=\left(a / n^{2}, b / n^{2}\right)\). 4 Let \(f\) be defined by \(f(x, y)=x^{2} y^{2} /\left(x^{2}+y^{2}\right)\), with \(f(0,0)=0\). By checking various sequences. test this for continuity at \((0,0)\). Can you tell whether or not it is continuous there?

Find \(\lim _{x \rightarrow 1} \frac{\frac{2}{x-1}+3}{4+\frac{5}{x^{2}-\frac{3}{x+2}}}\)

Let \(f\) and \(g\) each be uniformly continuous on a set \(E\). Show that \(f+g\) is uniformly continuous on \(E\).

Let \(f(x)=\left\\{\begin{array}{l}0 \text { if } x \text { is irrational } \\\ 1 / q \text { if } x \text { is the rational number } p / q \text { in lowest terms. }\end{array}\right.\) Is \(f\) continuous anywhere?

Let \(f\) be defined by \(f(x, y)=x^{2} y^{2} /\left(x^{2}+y^{2}\right)\), with \(f(0,0)=0\). By checking various sequences. test this for continuity at \((0,0)\). Can you tell whether or not it is continuous there?

Find \(\lim _{x \rightarrow 2} f(x)\) where \(f(x)=\frac{x^{3}+x^{2}-7 x+2}{2 x^{3}-5 x^{2}+6 x-8}\)

Let \(A\) and \(B\) be disjoint sets, and let \(f\) be continuous on \(A\) and continuous on \(B\). When is it continuous on \(A \cup B\) ?

Find inverses for the function \(f\) given by $$ f(x)=x^{2}-2 x-3 $$

For each of the following functions, find all the discontinuities and indicate any that are removable. (a) \(f(x)=\frac{x}{\sin (5 \cos x)}\) (b) \(f(x)=x+\frac{\sin x}{2 x-\frac{4}{x-1}}\) (c) \(f(x)=\frac{1}{1+e^{\sec x}}\)

Show by an example that the graph of a function defined on the interval \(0 \leq x \leq 1\) can be a closed set, without the function \(f\) being continuous.

Show that a real-valued function \(f\) is continuous in \(D\) if the set \(S=\\{\) all \(p \in D\) with \(b

Let \(A\) and \(B\) be disjoint closed sets and suppose that \(f\) is uniformly continuous on each. (a) Show that \(f\) is necessarily uniformly continuous on \(A \cup B\) if \(A\) is compact. (b) Show that \(f\) need not be uniformly continuous on \(A \cup B\) if neither \(A\) nor \(B\) is compact.

How many continuous inverses are there for the function described by $$ F(x)=x^{3}+3 x $$

For each of the following functions, find all the discontinuities and indicate any that are removable. (a) \(F(x, y)=\frac{x+2 y}{\sin (x+y)-\cos (x-y)}\) (b) \(F(x, y)=x \sin \left(\frac{y}{x}\right)\)

Let \(f\) and \(g\) be continuous on the interval \([0,1]\), and suppose that \(f(x)=g(x)\) for every rational number \(x=a / b\) in this interval. Prove that \(f=g\).

If \(f\) is uniformly continuous on \(D\), show that it has the property that if \(p_{n}, q_{n} \in D\) and \(\left|p_{n}-q_{n}\right| \rightarrow 0\), then \(\left|f\left(p_{n}\right)-f\left(q_{n}\right)\right| \rightarrow 0\) 7 Let \(D\) be a bounded set and let \(f\) be uniformly continuous on \(D \subset \mathbf{R}^{n} .\) Prove that \(f\) is bounded on \(D\).

Is there any interval on which the function \(f\) described by $$ f(x)=2 x+|x|-|x+1| $$ fails to have an inverse?

Investigate the behavior of \(F(x, y)\) at \((0,0)\) if (a) \(F(x, y)=\frac{x^{2} y}{2 x^{2}+y^{2}}\) (b) \(F(x, y)=\frac{x^{2} y}{3 x^{4}+2 y^{2}}\)

Use the intermediate value theorem to prove that any polynomial of odd degree with real coefficients has at least one real root.

Use the example \(f(x, y)=x^{2}\) to show that a continuous function does not have to map an open set onto an open set.

Let \(D\) be a bounded set and let \(f\) be uniformly continuous on \(D \subset \mathbf{R}^{n} .\) Prove that \(f\) is bounded on \(D\).

Show that a continuous function \(f\) cannot map the interval \([0,1]\) onto itself exactly 2 -to-1.

Let \(f\) and \(g\) be continuous on \([0,1]\) and suppose that \(f(0)g(1)\). Prove that there is a point \(x, 0

Use the example \(f(x)=x^{2} /\left(1+x^{2}\right)\) to show that a continuous function does not always have to map a closed set onto a closed set.

Let \(f\) be a function defined on a set \(E\) which is such that it can be uniformly approximated within \(\varepsilon\) on \(E\) by functions \(F\) that are uniformly continuous on \(E\), for every \(\varepsilon>0\). Show that \(f\) must itself be uniformly continuous on \(E\).

Investigate the existence of local and global inverses for the function \(f(x)=A x-\sin x\), for various values of \(A\).

The function $$ \exp \left(\frac{x^{2}+y^{2}-x y}{x^{2}+y^{2}}\right)=f(x, y) $$ is continuous on the open first quadrant. (a) Is it bounded there? (b) Can \(f\) be extended continuously to the closed first quadrant?

Show that any function that is locally constant on an open connected set \(D\) is in fact constant on \(D\).

Using the assumed continuity properties of the sine function, what can you say about the set on which the function \(g(x)=\csc (\sin (1 / x))\) is continuous?

Let \(f\) be a continuous function defined on the interval \(I=[a, b]\) which maps \(I\) onto \(I 1\) -to- 1 and which is its own inverse. (a) Show that, except for one possible function, \(f\) must be monotonic decreasing on \(I\). (b) What are the polynomial functions that are choices for \(f ?\)

Let \(f\) be an increasing function on the interval \([0,1] .\) Show that \(f\) cannot have more than a countable number of discontinuities on this interval. (Hint: First look at the one-sided limits at a point \(x_{0} .\) )

Prove that a set \(S\) is disconnected if and only if there is a real-valued function \(f\) that is continuous on \(S\) but takes only the values 2 and 3 on \(S\).

Show that \(f\) is continuous if and only if the inverse images of closed sets are closed sets relative to \(D\).

Let \(p_{1}, p_{2}, p_{3}\) be the vertices of a triangle \(D\) in the plane. (a) Show that any point \(p\) in \(D\) can be expressed as $$ p=\alpha_{1} p_{1}+\alpha_{2} p_{2}+\alpha_{3} p_{3} \quad \text { where } \alpha_{i} \geq 0 \text { and } \alpha_{1}+\alpha_{2}+\alpha_{3}=1 $$ (b) If \(f\) is a function defined on \(D\) with Lipschitz constant \(M\), use part \((a)\) to define a function \(F\) on \(D\) by $$ F(p)=\alpha_{1} f\left(p_{1}\right)+\alpha_{2} f\left(p_{2}\right)+\alpha_{3} f\left(p_{3}\right) $$ and show that \(\|F-f\|_{D} \leq M \operatorname{diam}(D)\).

Give a mathematical argument to show that a heated wire in the shape of a circle must always have two diametrically opposite points with the same temperature.

Let \(f\) be continuous on the interval \([a, b] .\) Given \(\varepsilon>0\) and a point \(t\) in the interval, choose \(\rho=\rho(t)\) so that, if \(|x-t|<\rho\), then \(|f(x)-f(t)|<\varepsilon\). Let \(U_{t}\) be the symmetric interval centered on \(t\) of radius \(\frac{1}{2} \rho(t) .\) Show that there are points \(t_{1}, t_{2}, \cdots, t_{m}\) such that the sets \(U_{t,}\) together cover the interval \([a, b]\).

How are \(f^{-1}(A \cap B)\) and \(f^{-1}(A \cup B)\) related to \(f^{-1}(A)\) and \(f^{-1}(B)\) ?

Let \(f: X \rightarrow Y\) be a function, and \(S\) and \(T\) arbitrary sets. Show that: (a) \(f f^{-1}(S) \subset S\) (b) \(T \subset f^{-1} f(T)\)

(a) Show that any heated tetrahedron must have three points located on its edges or vertices that have the same temperature. (b) Can you prove there must actually be four such points with the same temperature?

Formulate the definition of continuity for a complex-valued function \(f\). Show that \(f\) is continuous if and only if its real and imaginary parts are continuous functions.

Suppose that \(f\) is continuous on \([a, b]\) and that \(f(a) f(b)<0\). Prove Theorem 14 by filling in the details of the following argument. (a) Apply the process of repeated bisection to construct two sequences \(\left\\{a_{n}\right\\}\) and \(\left\\{b_{n}\right\\}\) such that \(a \leq a_{n}