Showing codimension of subspace of C[0,1] equals 1

Question

Show that $\overline{\{f∈C^1[0,1]:f(0)=0\}}$ as a subspace of $C[0,1]$ has codimension 1.

Attempt: define $T:C[0,1]\to$ $\Bbb{R}$ by $T(f)=f(0)$.
$T$ is a surjective continuous linear transfomation between Banach spaces.
Thus $C[0,1]/\{f∈C[0,1]:f(0)=0\}\cong\Bbb{R}$ and $\mathrm{codim}(\{f∈C[0,1]:f(0)=0\})=1$.
As $\overline{\{f∈C^1[0,1]:f(0)=0\}}\subseteq\{f∈C[0,1]:f(0)=0\}$
it follows that $\mathrm{codim}\overline{\{f∈C^1[0,1]:f(0)=0\}}\ge$$1$
I don't know how to show the other direction of the inequality.
Thank you.

Answer 1

Eventually I've shown that $\ker(T)=\overline{\{f∈C^1[0,1]:f(0)=0\}}$.
By Weierstrass approximation theorem I've a polynomial function $p_{1}$ s.t. $||f-p_{1}||<\mathcal{E}/2$.
Let $p(x)=p_{1}(x)-p_{1}(0)$. Then $p\in\overline{\{f∈C^1[0,1]:f(0)=0\}}$ and $||f-p||<\mathcal{E}$ as desired.