# Definition:Continuous Function

## Definition

### Continuous Complex Function

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

### 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 $A$.

### Definition by Epsilon-Delta

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

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

### Definition by Neighborhood

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

$\displaystyle \lim_{y \mathop \to x} \, \map f y = \map f x$
for every $\epsilon$-neighborhood $N_\epsilon$ of $\map f x$ in $\R$, there exists a $\delta$-neighborhood $N_\delta$ of $x$ in $A$ such that $\map f x \in N_\epsilon$ whenever $x \in N_\delta$.

### 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 $A$.