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.



My favorite counterexample in the real case works just as well for the complex case. $\frac{\sin nx} n$ converges uniformly to $0$ and is a sequence of analytic functions. The sequence of derivatives does not converge, uniformly or otherwise.


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