I came across the statement below:
Let $C([0,1])$ be the space of all continuous functions over the interval $[0,1]$ equipped with the Supremum norm. Assume $A$ is a map on the space of all differentiable functions whose derivative is continuous into $C([0,1])$. Also, $A$ is differentiation in the sense that it maps a functions to its derivative. The map $A$ (differentiation) is discontinuous.
It's written that the last sentence is well-known but I can't make any sense of it. How can I arrive at such a conclusion? Actually, I am looking for an explicit counterexample.
Any help would be highly appreciated.