dense subspace of $C(0,T)$

Question

I want to prove that the space H of functions which are continuous in [0,T] with weak derivative in $L^2[0,T]$ and their value in 0 is 0, is dense in the space of continuous functions in [0,T] with $f(0)=0$. I know that the functions $C^\infty(R)|_{[0,T]}$ are dense in H. But how can I approximate a continuous function ($f(0)=0$) with functions in $C^\infty(R)|_{[0,T]}$ because the convolution doesn't preserve the fact that $f(0)=0$. I would appreciate it if anyone could give me any idea. Thank you in advance.