Definition:Discontinuity (Real Analysis)/Jump

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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 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

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