Let $X$ be the closed unit disk in the complex plane. Let's consider $D^1 (X)$ to be the set of all functions $f: X \to \mathbb C$ that are continuous in $X$ and have continuous derivative.
My question is:
Is $D: D^1 (X) \to C(X), \, D(f) = f'$, a continuous operator with the supremum norm in $D^1(X)$ and $C(X)$?
This is not true in the real case. However, in the complex case, we have some nice properties. However, I'm not so sure that this is true.