Category:Discontinuities of the First Kind

This category contains results about Discontinuities of the First Kind.
Definitions specific to this category can be found in Definitions/Discontinuities of the First Kind.

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.