In my text there's a problem which reads as:
Consider $C[0, 1]$ with the norm $\|.\|_\infty$. Let $Y$ be the vector subspace of all differentiable functions on $[0, 1].$ Consider the linear map $D: (Y, \|.\|_\infty)\to(C[0, 1],\|.\|_\infty)$ given by $Df = f'$, the derivative of $f$. Show that $D$ is not continuous.
Is $D$ well-defined for what is the guarantee of the continuity of the derived function on $[0,1]$ of a continuous function on $[0,1]?$