Definition:Discontinuity (Real Analysis)/First Kind

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 $c \in X$.

Definition 1

$c$ is known as a discontinuity of the first kind 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

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.

Definition 2

$c$ is known as a discontinuity of the first kind of $f$ if and only if either:

$c$ is a jump discontinuity

or:

$c$ is a removable discontinuity.

Also known as

A discontinuity of the first kind is also known as a simple discontinuity.

Some authors define a discontinuity of the first kind and a jump discontinuity to be the same thing.

Other authors allow removable discontinuities to be a subset of jump discontinuities.

Also see

• Equivalence of Definitions of Discontinuity of the First Kind

• Definition:Discontinuity of the Second Kind

• Results about discontinuities of the first kind can be found here.