Combination Theorem for Continuous Functions/Complex/Difference Rule
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Theorem
Let $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions which are continuous on an open subset $S \subseteq \C$.
Then:
- $f - g$ is continuous on $S$.
Proof
We have that:
- $\map {\paren {f - g} } x = \map {\paren {f + \paren {-g} } } x$
From Multiple Rule for Continuous Complex Functions:
- $-g$ is continuous on $S$.
From Sum Rule for Continuous Complex Functions:
- $f + \paren {-g}$ is continuous on $S$.
The result follows.
$\blacksquare$