# Definition:Discontinuity (Real Analysis)/First Kind

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## Definition

Let $X$ be an open subset of $\R$.

Let $f: X \to Y$ be a real function.

Let $f$ be discontinuous at some point $c \in X$.

### Definition 1

$c$ is known as a **discontinuity of the first kind of $f$ if and only if:**

- $\ds \lim_{x \mathop \to c^-} \map f x$ and $\ds \lim_{x \mathop \to c^+} \map f x$ exist

where $\ds \lim_{x \mathop \to c^-} \map f x$ and $\ds \lim_{x \mathop \to c^+} \map f x$ denote the limit from the left and limit from the right at $c$ respectively.

### Definition 2

$c$ is known as a **discontinuity of the first kind** of $f$ if and only if either:

- $c$ is a
**jump discontinuity**

or:

- $c$ is a
**removable discontinuity**.

## Also known as

A **discontinuity of the first kind** is also known as a **simple discontinuity**.

Some authors define a **discontinuity of the first kind** and a **jump discontinuity** to be the same thing.

Some other authors allow **removable discontinuities** to be a subset of **jump discontinuities**.

## Also see

- Results about
**discontinuities of the first kind**can be found**here**.