Density of continuosly differentiable function in space of continuous functions with boundary conditions

Question

Let $A = \{f \in C([0,1]) \mid f(0) = 0 \}$ and let $B = \{f\in C^1([0,1])\mid f(0) = 0\}$. Then, is $B$ dense in $A$ with respect to the supremum norm?

I know that if the sets $A$ and $B$ do not have the condition $f(0)=0$, then the density follows from Weistrass's Theorem, but how do I prove/disprove the density in the case above?

Answer 1

Let $f\in C([01])$ with $f(0)=0$ and $c>0$, Stone Weirstrass implies there exists $g\in C^1([0,1])$ (a polynomial function) such that $\|f-g\|