# Definition:Discontinuous Mapping

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*This page is about Discontinuous Mapping. For other uses, see Discontinuity.*

## Definition

### Discontinuous Real Function

Let $f$ be a real function.

Then $f$ is **discontinuous** if and only if there exists at least one $a \in \R$ at which $f$ is discontinuous.

### Discontinuous Topological Space

Let $T_1 = \struct {A_1, \tau_1}$ and $T_2 = \struct {A_2, \tau_2}$ be topological spaces.

Let $f: A_1 \to A_2$ $x \in T_1$ be a mapping from $A_1$ to $A_2$.

Then by definition $f$ is continuous at $x$ if for every neighborhood $N$ of $\map f x$ there exists a neighborhood $M$ of $x$ such that $f \sqbrk M \subseteq N$.

Therefore, $f$ is **discontinuous at $x$** if for some neighborhood $N$ of $\map f x$ and every neighborhood $M$ of $x$:

- $f \sqbrk M \nsubseteq N$

The point $x$ is called a **discontinuity of $f$**.

## Also see

- Results about
**discontinuous mappings**can be found**here**.