Question

Show that the function f(x) =(1/x) is uniformly continuous on [2,∞). Then, show that f is not uniformly continuous on (0,∞)

Answer 1

Answer:Hence shown

Step-by-step explanation:

Given

$F\left ( x\right )=\frac{1}{x}$

We have to show the function is uniformly continuous in interval [2,∞)

A function is said to be continuous in given interval if it is differentiable in that interval

so derivative of $\frac{1}{x}$ is

$\frac{\mathrm{d} \frac{1}{x}}{\mathrm{d} x}$

${F}'(x)=-\frac{1}{x^2}$

${F}'(x) is\ valid\ for\ al\l values\ of\ x\ except\ x=0$

therefore $\frac{1}{x}$ is continuous in [2,∞) and not uniformly continuous in (0,∞] because ${F}'(x)=-\frac{1}{x^2}$ is inderminate at x=0 as $\frac{1}{0}$ is not defined.