Category:Combination Theorems for Continuous Functions
This category contains pages concerning Combination Theorem for Continuous Functions:
Real Functions
Let $\R$ denote the real numbers.
Let $f$ and $g$ be real functions which are continuous on an open subset $S \subseteq \R$.
Let $\lambda, \mu \in \R$ be arbitrary real numbers.
Then the following results hold:
Sum Rule
- $f + g$ is continuous on $S$.
Difference Rule
- $f - g$ is continuous on $S$.
Multiple Rule
- $\lambda f$ is continuous on $S$.
Combined Sum Rule
- $\lambda f + \mu g$ is continuous on $S$.
Product Rule
- $f g$ is continuous on $S$
Quotient Rule
- $\dfrac f g$ 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 $\C$ denote the complex numbers.
Let $f$ and $g$ be complex functions which are continuous on an open subset $S \subseteq \C$.
Let $\lambda, \mu \in \C$ be arbitrary complex numbers.
Then the following results hold:
Sum Rule
- $f + g$ is continuous on $S$.
Multiple Rule
- $\lambda f$ is continuous on $S$.
Combined Sum Rule
- $\lambda f + \mu g$ is continuous on $S$.
Product Rule
- $f g$ is continuous on $S$
Quotient Rule
- $\dfrac f g$ is continuous on $S \setminus \set {z \in S: \map g z = 0}$
that is, on all the points $z$ of $S$ where $\map g z \ne 0$.
Subcategories
This category has only the following subcategory.
Pages in category "Combination Theorems for Continuous Functions"
The following 26 pages are in this category, out of 26 total.
C
- Combination Theorem for Continuous Complex Functions
- Combination Theorem for Continuous Functions
- Combination Theorem for Continuous Functions/Combined Sum Rule
- Combination Theorem for Continuous Functions/Complex
- Combination Theorem for Continuous Functions/Complex/Combined Sum Rule
- Combination Theorem for Continuous Functions/Complex/Difference Rule
- Combination Theorem for Continuous Functions/Complex/Multiple Rule
- Combination Theorem for Continuous Functions/Complex/Product Rule
- Combination Theorem for Continuous Functions/Complex/Quotient Rule
- Combination Theorem for Continuous Functions/Complex/Sum Rule
- Combination Theorem for Continuous Functions/Difference Rule
- Combination Theorem for Continuous Functions/Multiple Rule
- Combination Theorem for Continuous Functions/Product Rule
- Combination Theorem for Continuous Functions/Quotient Rule
- Combination Theorem for Continuous Functions/Real
- Combination Theorem for Continuous Functions/Sum Rule
- Combination Theorem for Continuous Real Functions
- Combined Sum Rule for Continuous Complex Functions