## Proposition

Erdös contends that in Complex Analysis, versus Single Variable Calculus, a function will not have a derivative of zero when it reaches a maximum value.

In Single Variable Calculus:

A function $$f(x)$$, which is smooth (continuous and differentiable) on a open domain $$D$$, will have a derivative of zero when it attains a maximum on $$D$$.

However, for a complex function $$g(z)$$, on a closed domain $$U$$ with border $$B$$, the maximum (Guaranteed by the Extreme Value Theorem), may not have a derivative of zero.

## Proof

Let there exist a complex $$\epsilon$$ such that $$|\epsilon| > 0$$ and $$\delta$$ such that $$|\delta| > 0$$ where $$\epsilon$$ and $$\delta$$ are arbitrarily small.

Furthermore, let the complex function $$f(z)$$ be analytic and non-constant on an open domain $$D$$, bounded by a contour $$B$$ along which $$f(z)$$ is continuous and differentiable. Thus, $$f$$ is continuous and differentiable on the closed region $$U = D \cup B$$.

By the Maximum Modulus Theorem, the maximum of $$f$$ occurs on $$B$$ at $$z_0$$.

Now, suppose that $$f'(z_0)$$ is indeed zero. This would mean that for some $$\epsilon$$, there is are points $$w = z_0 + \delta$$ where $$\delta < \epsilon$$ for which $$f(w)=f(z_0)$$, since $$f(z)$$ is continuous. $$z_0 + \delta$$ is a maximum point for $$f(z)$$ on $$D$$. By the Maximum Modulus Theorem, this means that $$f$$ must be constant on $$D$$.

Since $$f(z)$$ is non-constant on $$D$$, this is a contradiction. $$#$$