# Definition:Discontinuity of the 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 $c \in X$.

Then $c$ is called a **discontinuity of the first kind** of $f$ if and only if:

- $\displaystyle \lim_{x \mathop \to c^-} f \left({x}\right)$ and $\displaystyle \lim_{x \mathop \to c^+} f \left({x}\right)$ exist

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

## Also known as

Some authors take these **discontinuities of the first kind** and jump discontinuities to be synonymous.

The difference is that some authors allow removable discontinuities to be a subset of jump discontinuities. Other authors choose to differentiate the two concepts.

## Sources

- 1953: Walter Rudin:
*Principles of Mathematical Analysis*: $4.5$