Chapter 3

Define \(f(x)=x^{2}\) for all \(x\). Verify the \(\epsilon-\delta\) criterion for continuity at \(x=2\) and at \(x=50\)

For each of the following statements, determine whether it is true or false and justify your answer. a. If the function \(f+g: \mathbb{R} \rightarrow \mathbb{R}\) is continuous, then the functions \(f: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) also are continuous. b. If the function \(f^{2}: \mathbb{R} \rightarrow \mathbb{R}\) is continuous, then so is the function \(f: \mathbb{R} \rightarrow \mathbb{R}\). c. If the functions \(f+g: \mathbb{R} \rightarrow \mathbb{R}\) and \(g: \mathbb{R} \rightarrow \mathbb{R}\) are continuous, then so is the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) d. Every function \(f: \mathbb{N} \rightarrow \mathbb{R}\) is continuous, where \(\mathbb{N}\) denotes the set of natural numbers.

Prove that a. \(\lim _{x \rightarrow 1} \frac{x^{4}-1}{x-1}=4\) b. \(\lim _{x \rightarrow 1} \frac{\sqrt{x}-1}{x-1}=\frac{1}{2}\)

If the function \(f: D \rightarrow \mathbb{R}\) is uniformly continuous and \(\alpha\) is any number, show that the function \(\alpha f: D \rightarrow \mathbb{R}\) also is uniformly continuous.

Define \(f(x)=\sqrt{x}\) for all \(x \geq 0 .\) Verify the \(\epsilon-\delta\) criterion for continuity at \(x=4\) and at \(x=100 .\) Hint: First show that for \(x \geq 0, x_{0}>0\) $$ \left|\sqrt{x}-\sqrt{x_{0}}\right| \leq\left|x-x_{0}\right| / \sqrt{x_{0}} $$

a. Find a continuous function \(f:(0,1) \rightarrow \mathbb{R}\) with an image equal to \(\mathbb{R}\). b. Find a continuous function \(f:(0,1) \rightarrow \mathbb{R}\) with an image equal to [0,1] c. Find a continuous function \(f: \mathbb{R} \rightarrow \mathbb{R}\) that is strictly increasing and has an image equal to (-1,1)

Find a maximizer for each of the following functions. a. \(f:[0,1] \rightarrow \mathbb{R}\) defined by \(f(x)=\sqrt{x}+x^{10}+4\) for \(0 \leq x \leq 1\) b. \(g:[-1,1] \rightarrow \mathbb{R}\) defined by \(g(x)=-x^{10}(x-1 / 4)^{24}\) for \(-1 \leq x \leq 1\) c. \(h:[-1,1] \rightarrow \mathbb{R}\) defined by \(h(x)=4-2 x^{3}\) for \(-1 \leq x \leq 1\)

Define $$ f(x)=\left\\{\begin{array}{ll} 11 & \text { if } 0 \leq x \leq 1 \\ x & \text { if } 1

Prove that if \(f: D \rightarrow \mathbb{R}\) and \(g: D \rightarrow \mathbb{R}\) are uniformly continuous, then so is the \(\operatorname{sum} f+g: D \rightarrow \mathbb{R}\)

Define \(f(x)=x^{3}\) for all \(x\). Verify the \(\epsilon-\delta\) criterion for continuity at each point \(x_{0}\).

Find the images of each of the following functions: a. \(f:[0, \infty) \rightarrow \mathbb{R}\) defined by \(f(x)=1 /\left(1+x^{2}\right)\) for \(x \geq 0\). b. \(h:(0,1) \rightarrow \mathbb{R}\) defined by \(h(x)=1 /\left(x^{2}+8 x\right)\) for \(0

Prove that there is a solution of the equation $$ \frac{1}{\sqrt{x+x^{2}}}+x^{2}-2 x=0, \quad x>0 $$

Define $$ f(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } x \leq 0 \\ x+1 & \text { if } x>0 \end{array}\right. $$ At what points is the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) continuous? Justify your answer.

Let \(a\) and \(b\) be real numbers with \(a

Find the following limits or determine that they do not exist: a. \(\lim _{x \rightarrow 0} \frac{1+1 / x}{1+1 / x^{2}}\) b. \(\lim _{x \rightarrow 0} \frac{1+1 / x^{2}}{1+1 / x}\) c. \(\lim _{x \rightarrow 1} \frac{1+1 /(x-1)}{2+1 /(x-1)^{2}}\)

Define $$ f(x)=\left\\{\begin{array}{ll} x+1 & \text { if } x \leq 3 / 4 \\ 2 & \text { if } x>3 / 4 \end{array}\right. $$ Use the \(\epsilon-\delta\) criterion for continuity at a point to show that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is not continuous at \(x=3 / 4\)

Define $$f(x)=\left\\{\begin{array}{ll}x-1 & \text { if } x<0 \\\x+1 & \text { if } x \geq 0\end{array}\right.$$ Show that \(f: \mathbb{R} \rightarrow \mathbb{R}\) is strictly increasing and that \(f^{-1}: f(\mathbb{R}) \rightarrow \mathbb{R}\) is continuous at 1 .

For a function \(f: D \rightarrow \mathbb{R},\) a solution of the equation $$ f(x)=x, \quad x \text { in } D $$ is called a fixed point of \(f .\) A fixed point corresponds to a point at which the graph of the function \(f\) intersects the line \(y=x .\) If \(f:[-1,1] \rightarrow \mathbb{R}\) is continuous, \(f(-1)>-1,\) and \(f(1)<1,\) show that \(f:[-1,1] \rightarrow \mathbb{R}\) has a fixed point.

For a function \(f: D \rightarrow \mathbb{R}\) and a point \(x_{0}\) in \(D,\) define \(A=\left\\{x\right.\) in \(\left.D \mid x \geq x_{0}\right\\}\) and \(B=\left\\{x\right.\) in \(\left.D \mid x \leq x_{0}\right\\} .\) Prove that \(f: D \rightarrow \mathbb{R}\) is continuous at \(x_{0}\) if and only if \(f: A \rightarrow \mathbb{R}\) and \(f: B \rightarrow \mathbb{R}\) are continuous at \(x_{0}\)

Suppose that \(S\) is a nonempty set of real numbers that is not sequentially compact. Prove that either (i) there is an unbounded sequence in \(S\) or (ii) there is a sequence in \(S\) that converges to a point \(x_{0}\) that is not in \(S\).

Let \(D\) be the set of real numbers consisting of the single number \(x_{0} .\) Show that the set \(D\) has no limit points. Also show that the set \(\mathbb{N}\) of natural numbers has no limit points.

Suppose that the functions \(h:[a, b] \rightarrow \mathbb{R}\) and \(g:[a, b] \rightarrow \mathbb{R}\) are continuous. Observe that a solution of the equation $$ h(x)=g(x), \quad x \text { in }[a, b] $$ corresponds to a point where the graphs intersect. Show that if \(h(a) \leq g(a)\) and \(h(b) \geq g(b),\) then this equation has a solution.

If a set \(S\) contains an unbounded sequence, show that the function \(f: S \rightarrow \mathbb{R}\) defined by \(f(x)=x\) for all \(x\) in \(S,\) is continuous but unbounded. If a set \(S\) contains a sequence that converges to a point \(x_{0}\) not in \(S,\) show that the function \(f: S \rightarrow \mathbb{R}\) defined by \(f(x)=1 /\left|x-x_{0}\right|\) for all \(x\) in \(S,\) is continuous but unbounded.

Show that it is not necessarily the case that if \(f: D \rightarrow \mathbb{R}\) and \(g: D \rightarrow \mathbb{R}\) are each uniformly continuous, then so is the product \(f g: D \rightarrow \mathbb{R}\)

Define the function \(g: \mathbb{R} \rightarrow \mathbb{R}\) by $$ g(x)=\left\\{\begin{array}{ll} x^{2} & \text { if } x \text { is rational } \\ -x^{2} & \text { if } x \text { is irrational. } \end{array}\right. $$ At what points is the function continuous? Justify your answer.

Suppose that the functions \(f: D \rightarrow \mathbb{R}\) and \(g: D \rightarrow \mathbb{R}\) are uniformly continuous and bounded. Prove that the product \(f g: D \rightarrow \mathbb{R}\) also is uniformly continuous. Hint: Write $$ f(u) g(u)-f(v) g(v)=f(u)[g(u)-g(v)]+g(v)[f(u)-f(v)] $$

For an odd natural number \(n,\) define \(f(x)=x^{n}\) for all \(x .\) Prove that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is strictly increasing and \(f(\mathbb{R})=\mathbb{R}\).

Define \(f(x)=\sqrt{x}\) for \(0 \leq x \leq 1\) a. Prove that the function \(f:[0,1] \rightarrow \mathbb{R}\) is continuous. b. Use part (a) to show that \(f:[0,1] \rightarrow \mathbb{R}\) is uniformly continuous. c. Show that \(f:[0,1] \rightarrow \mathbb{R}\) is not a Lipschitz function.

Suppose that the function \(f:[0,1] \rightarrow \mathbb{R}\) is continuous, \(f(0)>0,\) and \(f(1)=0\) Prove that there is a number \(x_{0}\) in (0,1] such that \(f\left(x_{0}\right)=0\) and \(f(x)>0\) for \(0 \leq x

For an unbounded nonempty set of real numbers \(D\), does there necessarily exist a continuous function \(f: D \rightarrow \mathbb{R}\) that is not uniformly continuous?

Suppose the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) has the property that there is some \(M>0\) such that $$ |f(x)| \leq M|x|^{2} $$ for all \(x\). Prove that $$ \lim _{x \rightarrow 0} f(x)=0 $$ and $$ \lim _{x \rightarrow 0} \frac{f(x)}{x}=0. $$

Let \(p: \mathbb{R} \rightarrow \mathbb{R}\) be a polynomial of odd degree. Prove that there is a solution of the equation $$ p(x)=0, \quad x \text { in } \mathbb{R} $$

Define the function \(h:[1,2] \rightarrow \mathbb{R}\) as follows: \(h(x)=0\) if the point \(x\) in [1,2] is irrational; \(h(x)=1 / n\) if the point \(x\) in [1,2] is rational and \(x=m / n,\) where \(m\) and \(n\) are natural numbers having no common positive integer factor other than \(1 .\) a. Prove that \(h:[1,2] \rightarrow \mathbb{R}\) fails to be continuous at each rational number in [1,2] . b. Prove that if \(\epsilon>0,\) then the set \(\\{x\) in \([1,2] \mid h(x)>\epsilon\\}\) has only a finite number of points. c. Use part (b) to prove that \(h:[1,2] \rightarrow \mathbb{R}\) is continuous at each irrational number in [1,2]

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuous at the point \(x_{0}\) and that \(f\left(x_{0}\right)>0\). Prove that there is an interval \(I \equiv\left(x_{0}-1 / n, x_{0}+1 / n\right),\) where \(n\) is a natural number, such that \(f(x)>0\) for all \(x\) in \(I\). (Hint: Argue by contradiction.)

For each number \(x\), define \(f(x)\) to be the largest integer that is less than or equal to \(x\). Graph the function \(f: \mathbb{R} \rightarrow \mathbb{R}\). Given a number \(x_{0}\), examine $$ \lim _{x \rightarrow x_{0}} f(x). $$

For positive numbers \(a\) and \(b\) and natural numbers \(n\) and \(m,\) show that \(a=b\) if and only if \(a^{n}=b^{n}\) if and only if \(a^{1 / m}=b^{1 / m}.\)

A function \(f: D \rightarrow \mathbb{R}\) is called a Lipschitz function if there is some nonnegative number \(C\) such that $$ |f(u)-f(v)| \leq C|u-v| $$ for all points \(u\) and \(v\) in \(D\). Prove that if \(f: D \rightarrow \mathbb{R}\) is a Lipschitz function, then it is uniformly continuous.

Let \(k\) be a natural number. Prove that $$ \lim _{x \rightarrow 1} \frac{x^{k}-1}{x-1}=k $$

Show that for a positive number \(x\) and integers \(m\) and \(n,\) with \(n\) positive, $$ \left(x^{1 / n}\right)^{m}=\left(x^{m}\right)^{1 / n} $$

Let \(I\) be a nonempty convex subset of \(\mathbb{R}\). If \(I\) is bounded above, define \(b=\sup I\); if \(I\) is bounded below, define \(a=\inf I\). Prove the following: a. If \(I\) is unbounded above and below, then \(I=\mathbb{R}\). b. If \(I\) is bounded below but not above, then \(I=(a, \infty)\) or \(I=[a, \infty)\). c. If \(I\) is bounded above but not below, then \(I=(-\infty, b]\) or \(I=(-\infty, b)\). d. If \(I\) is bounded, then \(I\) is one of the sets \([a, b],(a, b),[a, b),(a, b]\).

Suppose that the function \(g: \mathbb{R} \rightarrow \mathbb{R}\) is continuous and that \(g(x)=0\) if \(x\) is rational. Prove that \(g(x)=0\) for all \(x\) in \(\mathbb{R}\).

Suppose that the function \(f: D \rightarrow \mathbb{R}\) is not uniformly continuous. Then, by definition, there are sequences \(\left\\{s_{n}\right\\}\) and \(\left\\{t_{n}\right\\}\) in \(D\) such that $$ \lim _{n \rightarrow \infty}\left[s_{n}-t_{n}\right]=0, \text { but } \lim _{n \rightarrow \infty}\left[f\left(s_{n}\right)-f\left(t_{n}\right)\right] \neq 0 $$ a. Show that there is an \(\epsilon>0\) and a strictly increasing sequence of indices \(\left\\{n_{k}\right\\}\) such that for each index \(k,\left|f\left(s_{n_{k}}\right)-f\left(t_{n_{k}}\right)\right| \geq \epsilon\) b. Define \(u_{k}=s_{n_{k}}\) and \(v_{k}=t_{n_{k}}\) for each index \(k .\) Show that \(\lim _{n \rightarrow \infty}\left[u_{n}-v_{n}\right]=0,\) but \(\left|f\left(u_{n}\right)-f\left(v_{n}\right)\right| \geq \epsilon\) for each index \(n\)

(A General Monotone Convergence Principle.) Let \(a\) and \(b\) be numbers with \(a

Let the function \(f: D \rightarrow \mathbb{R}\) be continuous. Then define the function \(|f|: D \rightarrow \mathbb{R}\) by \(|f|(x)=|f(x)|\) for \(x\) in \(D .\) Prove that the function \(|f|: D \rightarrow \mathbb{R}\) also is continuous.

Let the function \(f:[a, b] \rightarrow \mathbb{R}\) be continuous and one-to- one and such that \(f(a)<\) \(f(b) .\) Let \(c\) be a point in the open interval \((a, b) .\) Prove that \(f(a)

Suppose that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) has the property that \(f(u+v)=f(u)+f(v) \quad\) for all \(u\) and \(v\) a. Define \(m \equiv f(1) .\) Prove that \(f(x)=m x \quad\) for all rational numbers \(x\) b. Use (a) to prove that if \(f: \mathbb{R} \rightarrow \mathbb{R}\) is continuous, then $$ f(x)=m x \quad \text { for all } x $$