Derivative of Constant/Complex

Theorem

Let $f_c \left({z}\right)$ be the constant function on an open domain $D \in \C$, where $c \in \C$.

Then:

$\forall z \in D : f_c' \left({z}\right) = 0$

The function $f_c: D \to \C$ is defined as:

$\forall z \in D: f_c \left({z}\right) = c$

Thus:

$\ds f_c' \left({z}\right)$

$=$

$\ds \lim_{h \mathop \to 0} \frac {f_c \left({z + h}\right) - f_c \left({z}\right)} h$

$\ds $

$=$

$\ds \lim_{h \mathop \to 0} \frac {c - c} h$

$\ds $

$=$

$\ds \lim_{h \mathop \to 0} \frac 0 h$

$\ds $

$=$

$\ds 0$

$\blacksquare$