Definition:Discontinuity (Real Analysis)/First Kind

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 $f$ :
 * $\displaystyle \lim_{x \mathop \to c^-} f \left({x}\right)$ 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.