Prove continuity using reverse triangle equality

Question

Given $$f(x) = \|x-a\|$$ prove using reverse triangle equality that this is a continuous function. So I proceed like this; we look at the equality $$| f(x) - f(b)|$$ and want to show that it's continuous on $b$. We thus get $$|\ ||x - a|| - ||b-a|| \ |$$ and using the RTE, I get $$| \ \|x - a\| - \|b-a\| \ | \le \|x-a -(b-a)\| = \|x - b\| $$.... does above make sense? And then what.... we'd set $\epsilon = ..... \delta$?

Answer 1

We have using the reverse triangle inequality:

$$\forall x,y\qquad|f(x)-f(y)|=\left|\|x-a\|-\|y-a\|\right|\le\|x-y\|$$ hence $f$ is lipschitzian function so it's continuous.

Answer 2

Now go on:

let $\epsilon > 0$. Take $\delta = \epsilon$. Then $$ \| x-b \| < \delta \implies |f(x) - f(b)| < \epsilon $$

This is a particular case of Lipschitz continuity.