# Is derivative operator of analytic functions continuous?

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?

## 1 Answer

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.