Category:Continuous Real-Valued Functions
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This category contains results about Continuous Real-Valued Functions.
Let $\R^n$ be the cartesian $n$-space.
Let $f: \R^n \to \R$ be a real-valued function on $\R^n$.
Then $f$ is continuous on $\R^n$ if and only if:
- $\forall a \in \R^n: \forall \epsilon \in \R_{>0}: \exists \delta \in \R_{>0}: \forall x \in \R^n: \map d {x, a} < \delta \implies \size {\map f x - \map f a} < \epsilon$
where $\map d {x, a}$ is the distance function on $\R^n$:
- $\ds d: \R^n \to \R: \map d {x, y} := \sqrt {\sum_{i \mathop = 1}^n \paren {x_i - y_i}^2}$
where $x = \tuple {x_1, x_2, \ldots, x_n}, y = \tuple {y_1, y_2, \ldots, y_n}$ are general elements of $\R^n$.
Subcategories
This category has the following 2 subcategories, out of 2 total.
Pages in category "Continuous Real-Valued Functions"
The following 9 pages are in this category, out of 9 total.
C
- Combination Theorem for Bounded Continuous Real-Valued Functions
- Continuous Real Function Bounded on Finite Subdivision
- Continuous Real Function is Bounded on Neighborhood of Argument
- Continuous Real-Valued Function is not necessarily Bounded
- Continuous Real-Valued Function on Compact Space is Bounded
- Continuous Real-Valued Function/Examples
- Continuous Real-Valued Function/Examples/Non-Continuous Example 1
- Continuous Real-Valued Function/Examples/Non-Continuous Example 2