Category:Definitions/Continuous Functions

This category contains definitions related to Continuous Functions.
Related results can be found in Category: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

$f$ is continuous at $x$ if and only if the limit $\ds \lim_{y \mathop \to x} \map f y$ exists and:

$\ds \lim_{y \mathop \to x} \map f y = \map f x$

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 $A$.