Continuity of $f(x) = \|x\|$

Question

Show that $f(x) = \|x\|$ is continuous.

I know that $f$ is continuous at $x'$ if $\forall\varepsilon >0$ there exists an $\delta$ such that $|f(x)-f(x')|< \varepsilon$.

So from here $|f(x)-f(x')| = \big| \|x\|- \|x'\|\big| \leq \|x-x'\|$.

However how do I find a good candidate for $\delta$ here since I cannot do a lot anymore with $||x-x'||$?

Answer 1

Take $\delta=\varepsilon$. Then,$$\|x-x'\|<\delta\iff\|x-x'\|<\varepsilon\implies\bigl|\|x\|-\|x'\|\bigr|<\varepsilon,$$since$$\bigl|\|x\|-\|x'\|\bigr|\leqslant\|x-x'\|.$$