Definition:Discontinuity of the First Kind/Definition 1
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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 $c \in X$.
$c$ 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.
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.
Some other authors allow removable discontinuities to be a subset of jump discontinuities.
Also see
- Results about discontinuities of the first kind can be found here.
Sources
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (next): $4.26$