Combination Theorem for Continuous Functions/Quotient Rule
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Theorem
Real Functions
Let $f$ and $g$ be real functions which are continuous on an open subset $S \subseteq \R$.
- $\dfrac f g$ is continuous on $S \setminus \set {x \in S: \map g x = 0}$
that is, on all the points $x$ of $S$ where $\map g x \ne 0$.
Complex Functions
Let $f$ and $g$ be complex functions which are continuous on an open subset $S \subseteq \C$.
- $\dfrac f g$ is continuous on $S \setminus \set {z \in S: \map g z = 0}$
that is, on all the points $z$ of $S$ where $\map g z \ne 0$.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): continuous function (continuous mapping)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): continuous function (continuous mapping, continuous map)