Non differentiable, continuous functions in metric spaces.

Your question is equivalent to asking

• Is it possible to approximate a continuous function uniformly with a differentiable one, or
• Is the set of differentiable function dense in $C([0,1])$.

Both is true and stays true if you replace "differentiable" by differentiable infintely often and can be seen by convolution with mollifiers.

That is true. In fact, one may even take the function $g$ to be a polynomial. See the Stone-Weierstrass approximation theorem.