Category:Continuous Functions

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This category contains results about Continuous Functions.


Continuous Complex Function

As the complex plane is a metric space, the same definition of continuity applies to complex functions as to metric spaces.


Continuous Real Function

Continuity at a Point

Let $A \subseteq \R$ be any subset of the real numbers.

Let $f: A \to \R$ be a real function.

Let $x \in A$ be a point of $A$.


Then $f$ is continuous at $x$ if and only if the limit $\displaystyle \lim_{y \to x} f \left({y}\right)$ exists and:

$\displaystyle \lim_{y \to x} \ f \left({y}\right) = f \left({x}\right)$


Continuous Everywhere

Let $f : \R \to \R$ be a real function.


Then $f$ is everywhere continuous if and only if $f$ is continuous at every point in $\R$.


Continuity on a Subset of Domain

Let $A \subseteq \R$ be any subset of the real numbers.

Let $f: A \to \R$ be a real function.


Then $f$ is continuous on $A$ if and only if $f$ is continuous at every point of $A$.

Subcategories

This category has only the following subcategory.

C

Pages in category "Continuous Functions"

The following 53 pages are in this category, out of 53 total.