Derivative of Constant/Complex

From ProofWiki
Jump to navigation Jump to search

Theorem

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


Then:

$\forall z \in D : \map { {f_c}'} z = 0$


Proof

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

$\forall z \in D: \map {f_c} z = c$

Thus:

\(\ds \map { {f_c}'} z\) \(=\) \(\ds \lim_{h \mathop \to 0} \frac {\map {f_c} {z + h} - \map {f_c} z} h\) Definition of Derivative of Complex Function
\(\ds \) \(=\) \(\ds \lim_{h \mathop \to 0} \frac {c - c} h\)
\(\ds \) \(=\) \(\ds \lim_{h \mathop \to 0} \frac 0 h\)
\(\ds \) \(=\) \(\ds 0\)

$\blacksquare$