An application of the Closed Graph Theorem

Question

Let $T:L^2([0,1]) \to L^2([0,1])$ be a bounded linear map of Hilbert spaces such that if $f\in L^2([0,1])$ is continuous then so is $Tf$. Show that there is a positive constant C such that $$sup_{x\in[0,1] }|Tf(x)|\leq C sup_{x\in[0,1] }|f(x)|$$