--- title: "Classification of discontinuities" chunk: 1/4 source: "https://en.wikipedia.org/wiki/Classification_of_discontinuities" category: "reference" tags: "science, encyclopedia" date_saved: "2026-05-05T09:08:17.602531+00:00" instance: "kb-cron" --- While continuous functions are important in mathematics, not all functions are continuous. If a function is not continuous at a limit point (also called an "accumulation point" or "cluster point") of its domain, it has a discontinuity there. The set of all points of discontinuity of a function may be a discrete set, a dense set, or even the entire domain of the function. == Classification == For each of the following, consider a real valued function f {\displaystyle f} of a real variable x , {\displaystyle x,} defined in a neighborhood of the point x 0 {\displaystyle x_{0}} at which f {\displaystyle f} is discontinuous. === Removable discontinuity === Consider the piecewise function f ( x ) = { x 2 for x < 1 0 for x = 1 2 − x for x > 1 {\displaystyle f(x)={\begin{cases}x^{2}&{\text{ for }}x<1\\0&{\text{ for }}x=1\\2-x&{\text{ for }}x>1\end{cases}}} The point x 0 = 1 {\displaystyle x_{0}=1} is a removable discontinuity. For this kind of discontinuity: The one-sided limit from the negative direction: L − = lim x → x 0 − f ( x ) {\displaystyle L^{-}=\lim _{x\to x_{0}^{-}}f(x)} and the one-sided limit from the positive direction: L + = lim x → x 0 + f ( x ) {\displaystyle L^{+}=\lim _{x\to x_{0}^{+}}f(x)} at x 0 {\displaystyle x_{0}} both exist, are finite, and are equal to L = L − = L + . {\displaystyle L=L^{-}=L^{+}.} In other words, since the two one-sided limits exist and are equal, the limit L {\displaystyle L} of f ( x ) {\displaystyle f(x)} as x {\displaystyle x} approaches x 0 {\displaystyle x_{0}} exists and is equal to this same value. If the actual value of f ( x 0 ) {\displaystyle f\left(x_{0}\right)} is not equal to L , {\displaystyle L,} then x 0 {\displaystyle x_{0}} is called a removable discontinuity. This discontinuity can be removed to make f {\displaystyle f} continuous at x 0 , {\displaystyle x_{0},} or more precisely, the function g ( x ) = { f ( x ) x ≠ x 0 L x = x 0 {\displaystyle g(x)={\begin{cases}f(x)&x\neq x_{0}\\L&x=x_{0}\end{cases}}} is continuous at x = x 0 . {\displaystyle x=x_{0}.} The term removable discontinuity is sometimes broadened to include a removable singularity, in which the limits in both directions exist and are equal, while the function is undefined at the point x 0 . {\displaystyle x_{0}.} This use is an abuse of terminology because continuity and discontinuity of a function are concepts defined only for points in the function's domain. === Jump discontinuity === Consider the function f ( x ) = { x 2 for x < 1 0 for x = 1 2 − ( x − 1 ) 2 for x > 1 {\displaystyle f(x)={\begin{cases}x^{2}&{\mbox{ for }}x<1\\0&{\mbox{ for }}x=1\\2-(x-1)^{2}&{\mbox{ for }}x>1\end{cases}}} Then, the point x 0 = 1 {\displaystyle x_{0}=1} is a jump discontinuity. In this case, a single limit does not exist because the one-sided limits, L − {\displaystyle L^{-}} and L + {\displaystyle L^{+}} exist and are finite, but are not equal: since, L − ≠ L + , {\displaystyle L^{-}\neq L^{+},} the limit L {\displaystyle L} does not exist. Then, x 0 {\displaystyle x_{0}} is called a jump discontinuity, step discontinuity, or discontinuity of the first kind. For this type of discontinuity, the function f {\displaystyle f} may have any value at x 0 . {\displaystyle x_{0}.} === Essential discontinuity === For an essential discontinuity, at least one of the two one-sided limits does not exist in R {\displaystyle \mathbb {R} } . (Notice that one or both one-sided limits can be ± ∞ {\displaystyle \pm \infty } ). Consider the function f ( x ) = { sin ⁡ 5 x − 1 for x < 1 0 for x = 1 1 x − 1 for x > 1. {\displaystyle f(x)={\begin{cases}\sin {\frac {5}{x-1}}&{\text{ for }}x<1\\0&{\text{ for }}x=1\\{\frac {1}{x-1}}&{\text{ for }}x>1.\end{cases}}} Then, the point x 0 = 1 {\displaystyle x_{0}=1} is an essential discontinuity. In this example, both L − {\displaystyle L^{-}} and L + {\displaystyle L^{+}} do not exist in R {\displaystyle \mathbb {R} } , thus satisfying the condition of essential discontinuity. So x 0 {\displaystyle x_{0}} is an essential discontinuity, infinite discontinuity, or discontinuity of the second kind. (This is distinct from an essential singularity, which is often used when studying functions of complex variables). == Counting discontinuities of a function == Supposing that f {\displaystyle f} is a function defined on an interval I ⊆ R , {\displaystyle I\subseteq \mathbb {R} ,} we will denote by D {\displaystyle D} the set of all discontinuities of f {\displaystyle f} on I . {\displaystyle I.} By R {\displaystyle R} we will mean the set of all x 0 ∈ I {\displaystyle x_{0}\in I} such that f {\displaystyle f} has a removable discontinuity at x 0 . {\displaystyle x_{0}.} Analogously by J {\displaystyle J} we denote the set constituted by all x 0 ∈ I {\displaystyle x_{0}\in I} such that f {\displaystyle f} has a jump discontinuity at x 0 . {\displaystyle x_{0}.} The set of all x 0 ∈ I {\displaystyle x_{0}\in I} such that f {\displaystyle f} has an essential discontinuity at x 0 {\displaystyle x_{0}} will be denoted by E . {\displaystyle E.} Of course then D = R ∪ J ∪ E . {\displaystyle D=R\cup J\cup E.} The two following properties of the set D {\displaystyle D} are relevant in the literature.