# Definition:Discontinuous Mapping

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

This page has been identified as a candidate for refactoring of medium complexity.In particular: This transclusion is messed upUntil this has been finished, please leave
`{{Refactor}}` in the code.
Because of the underlying complexity of the work needed, it is recommended that you do not embark on a refactoring task until you have become familiar with the structural nature of pages of $\mathsf{Pr} \infty \mathsf{fWiki}$.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Refactor}}` from the code. |

## Definition

### Discontinuous Real Function

### At a Point

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

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

Let $a\in A$.

Then $f$ is **discontinuous** at $a$ if and only if $f$ is not continuous at $a$.

### Discontinuous Topological Space

### At a Point

Let $T_1 = \left({A_1, \tau_1}\right)$ and $T_2 = \left({A_2, \tau_2}\right)$ 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 $f \left({x}\right)$ there exists a neighborhood $M$ of $x$ such that $f \left({M}\right) \subseteq N$.

Therefore, $f$ is discontinuous at $x$ if for some neighbourhood $N$ of $f \left({x}\right)$ and every neighbourhood $M$ of $x$, $f \left({M}\right) \nsubseteq N$.

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