Answer to Question #145454 in Calculus for Dolly

Question

Show that the function f given by f(x)=|x| is continuous on R^n. [Hint: Consider |x-a|^2=(x-a).(x-a).]

Answer 1

Let $\epsilon>0$ be given. We want to find a $\delta=\delta(\epsilon) >0$ such that for any $x_0 \in \mathbb{R}$ if $|x-x_0|<\delta$ then, $|f(x)-f(x_0)|<\epsilon$

Now,

$|f(x)-f(x_0)|=||x|-|x_0|| \leq|x-x_0|<\delta=\epsilon$ if $\delta=\epsilon.$

$\implies |f(x)-f(x_0)|<\epsilon$

Hence, $f(x)=|x|$ is continuous on $\mathbb{R}$