Definition:Jump Discontinuity

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 jump discontinuity of $f$ if and only if:

$\displaystyle \lim_{x \mathop \to c^-} \map f x$ and $\displaystyle \lim_{x \mathop \to c^+} \map f x$ exist and are not equal

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

Jump

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

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

Let $f$ be discontinuous at some point $c \in X$ such that $c$ is a jump discontinuity of $f$.

The jump at $c$ is defined as:

$\displaystyle \lim_{x \mathop \to c^+} \map f x - \lim_{x \mathop \to c^-} \map f x$

Also known as

Some authors take 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 distinguish between the two concepts.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): Entry: jump discontinuity
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): Entry: discontinuity
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): Entry: discontinuity
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): Entry: jump discontinuity