Continuity of differential operator

Question

I'm trying to show that the functional: $T:C^1_{[0,1]} \to C^0_{[0,1]}$ defined by $Tf = f'$ is continuous, using the metric $d(f,g)=\sup_{x \in [0,1]} |f(x) - g(x)|$.

I start by taking $f,g$ with $d(f,g) < \delta $ for some $\delta > 0$. Then proceed by:

$$ d(Tf, Tg) = d(f', g') = \sup_{x \in [0,1]} |f'(x) - g'(x)| $$

I'm not quite sure how to use the information about the distance between $f,g$ to continue though

Answer 1

This is not true. Let $f_n(x)=\frac{x^n}n$. Then $\lim_{n\to\infty}f_n=0$, but it is not true that $\lim_{n\to\infty}T(f_n)=T(0)=0$.