Let $\mathbb{D}$ be the set of functions $f:[0,1]\to \mathbb{R}$ of class $C^1$ (differentiable with continuous derivative). Let $\mathcal{C}[0,1]$ be the set of continuous functions in $[0,1]$ ($\to \mathbb{R}$).
Let $d_1,d_2$ be the metrics given by:
$d_1(f,g)=\displaystyle\int_0^1 |f(x)-g(x)| dx$
and
$d_2(f,g)=\sup_{x\in[0,1]} |f(x)-g(x)|$
Let $D:\mathbb{D}\to \mathcal{C}[0,1]$ be such that $D(f)=f'$.
a) Is $D:(\mathbb{D},d_1) \to (\mathcal{C}[0,1],d_1)$ a continuous function?
b) Is $D:(\mathbb{D},d_2) \to (\mathcal{C}[0,1],d_2)$ a continuous function?
DISCLAIMER: $D:(\mathbb{D},d_1)$ represents the topologic space given by the metric topology induced by $d_1$ (and so for $d_2$).
Well, this is a bit spoiler, but I'm quite sure that in both cases $D$ is not continuous. The idea would be to think of continuity in the "$\varepsilon-\delta$" sense. Some functions $F,G$ could be very "close", and however $F'$ and $G'$ be very far away. It's not hard to imagine examples of what i'm saying. However, it's a bit difficult to describe the problem precisely.
Best Answer
Hint: in both cases, you can find a sequence of functions which stay close to $0$, but whose derivatives go further and further away from $0$.
Look at the functions of the form $x^{-p}$ for various $p>0$.
(In fact, you may see that both of these metrics are induced by norms and $D$ is linear, so $D$ is continuous iff it is bounded.)