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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.

Help?

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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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