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

$f$ 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 $A$.