91Ó°ÊÓ

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

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.

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.

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.

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 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 \(n\)