showing a function is continuous and differentiable

Hello,

I have a problem in hand. I'll pose the question and include my attempt. Please correct me wherever I am wrong.

Let f and g are functions that satisfy |f(x)| =< |g(x)| for all x and such that g is not the zero function. Give examples of what g could be to make sure that 1) f is continuous at 0 2) f is differentiable at 0.

1) I couldn't solve it. any help?

2) If I chose function |f(x)| =< |g(x)| = x^2, for f to be differentiable at 0,

f'(0) = lim x->0 [f(x) - f(0)]/[x-0] = lim x->0 f(x)/x = limx->0 x^2/x = 0.

But on 2, am I even allowed to assume f(0) = 0?

I can rephrase things if there's a confusion. Thanks.