Definition:Discontinuity (Real Analysis)/Jump
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:
- $\ds \lim_{x \mathop \to c^-} \map f x$ and $\ds \lim_{x \mathop \to c^+} \map f x$ exist and are not equal
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.
Note that $\map f c$ may equal either of these limits, or neither, or may not even be defined.
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:
- $\ds \lim_{x \mathop \to c^+} \map f x - \lim_{x \mathop \to c^-} \map f x$
Examples
Example 1
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = \begin {cases} 1 & : x < 2 \\ 2 & : x \ge 2 \end {cases}$
Then $f$ has a jump discontinuity at $x = 2$.
In this case, $\map f 2$ is defined, and equals the limit from the right.
Example 2
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = \begin {cases} 0 & : x > 1 \\ 1 & : x < 1 \\ \dfrac 1 2 & : x = 1 \end {cases}$
Then $f$ has a jump discontinuity at $x = 1$.
In this case, $\map f 1$ is defined, but equals neither limit.
Example 3
Let $f: \R \to \R$ be the real function defined as:
- $\forall x \in \R: \map f x = \begin {cases} -1 & : x < 0 \\ 1 & : x > 0 \\ \text {undefined} & : x = 0 \end {cases}$
Then $f$ has a jump discontinuity at $x = 1$.
In this case, $\map f 0$ is not defined.
Also see
- Results about jump discontinuities can be found here.
Sources
- 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): jump discontinuity
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): discontinuity
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): discontinuity
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): jump discontinuity