Continuity of functions - Real Analysis

I have a homework assignement due tomorrow morning and I have some idea on how to do this problem but I am not very sure.

Define functions f and g on [-1,1] by

Prove that f is continous at 0 and that g is not continuous at 0. Explain why these functions are continuous at every other point in [-1,1].

So I was thinking for f(x) just using the definition of continuity:

Choose

Ok this proves that f is continuous at 0 right??

I am not entirely sure how to prove g(x) isn't continuous but also that these functions are continuous at every other points!

Any suggestion??