Definition:Jump Discontinuity

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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:

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