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.
Also see
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