Proving continuity of $x^2$

Question

I need to prove the continuity for: $$f:\mathbb{R}\to\mathbb{R}, x\mapsto x^2$$ What I've got so far: $$|x-x_0|<\delta \Rightarrow |x^2-x_0^2|<\epsilon$$ $$|x^2-x_0^2|=|x+x_0||x-x_0|<|x+x_0|\delta=|x-x_0+2x_0|\delta\leq(|x-x_0|+|2x_0|)\delta=(\delta+|2x_0|)\delta$$ but how do I continue, I am stuck here.
Thanks in advance.

Answer 1

If you now had $(\delta+2x_0)\delta = \epsilon$, you'd be done. Since you are free to choose $\delta$ based on $x_0$ and $\epsilon$, manipulate this equation so it gives you a choice for $\delta$.