So far, I have this:

Proving $\displaystyle |f(x)-f(y)|$ is continuous:

Need to show: $\displaystyle \forall \ \epsilon>0 \exists \delta>0 s.t \ |y-x| < \delta \ \Rightarrow |f(x)-f(y)|< \epsilon$.

$\displaystyle |y-x|< \delta \Rightarrow |y-x|^2< \delta^2$

Therefore: $\displaystyle |f(x)-f(y)| \leq (y-x)^2< \delta^2$

Let $\displaystyle \delta^2=\epsilon$

Hence $\displaystyle |f(x)-f(y)| \leq (y-x)^2< \delta^2=\epsilon$ so continuity is proved.

However, i'm sure this helps but I can't see why. I've also tried expanding the right and left hand sides but that doesn't help either.

Does anyone have any hints?