91Ó°ÊÓ

Show that none of the following mappings \(f: X \rightarrow X\) have a fixed point and explain why the Contraction Mapping Principle is not contradicted: a. \(X=(0,1) \subseteq \mathbb{R}\) and \(f(x)=x / 2\) for \(x\) in \(X\) b. \(X=\mathbb{R}\) and \(f(x)=x+1\) for \(x\) in \(X\) c. \(X=\left\\{(x, y)\right.\) in \(\left.\mathbb{R}^{2} \mid x^{2}+y^{2}=1\right\\}\) and \(f(x, y)=(-y, x)\) for \((x, y)\) in \(X\)

Let \(X=C([0,1], \mathbb{R}) .\) Find \(d(f, g)\) for each of the following pairs of functions: a. \(f(x)=x\) and \(g(x)=\cos x\) for \(x\) in [0,1] b. \(f(x)=4 x^{3}\) and \(g(x)=6 x^{2}-3 x\) for \(x\) in [0,1]

Prove that a metric space \(X\) is disconnected if and only if there is a subset \(D\) of \(X\) that is both open and closed in \(X,\) with \(D \neq \emptyset\) and \(D \neq X\).

Suppose that \(X\) is a metric space that contains the point \(p\) and \(r\) is a positive number. Prove that the set \(\\{q\) in \(X \mid d(p, q) \leq r\\}\) is closed in \(X\).

a. Define \(f(x)=\sqrt{x}\) for \(x \geq 0 .\) Show that the function \(f:[0, \infty) \rightarrow \mathbb{R}\) is continuous but is not Lipschitz. b. Define \(f(x)=|x|\) for all real numbers \(x .\) Show that the function \(f: \mathbb{R} \rightarrow \mathbb{R}\) is Lipschitz but not differentiable.

For any two points in the plane \(\mathbb{R}^{2},\) define $$ d^{*}(p, q)=\left|p_{1}-q_{1}\right|+\left|p_{2}-q_{2}\right| $$ a. Show that \(d^{*}\) defines a metric on \(\mathbb{R}^{2}\). b. Compare an open ball about (0,0) in this metric with an open ball about (0,0) in the Euclidean metric. c. Show that a sequence in \(\mathbb{R}^{2}\) converges with respect to the above metric if and only if it converges with respect to the Euclidean metric.

Let \(X\) be any set considered a metric space with the discrete metric. With this metric, show that every subset of \(X\) is both open and closed in \(X\).

Suppose that \(X\) is a set consisting of more than one point, considered a metric space with the discrete metric. Show that \(X\) is not connected.