Continuous functions are of utmost importance in mathematics and applications. However, not all functions are continuous. If a function is not continuous at a point in its domain, one says that 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.
This article describes the classification of discontinuities in the simplest case of functions of a single real variable taking real values.
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Consider a function ƒ of real variable x with real values defined in a neighborhood of a point x_{0}. Then three situations are possible in which the function ƒ is discontinuous at the point on the real axis x = x_{0}:
The term removable discontinuity is sometimes incorrectly used for cases in which the limits in both directions exist and are equal, while the function is undefined at the point x_{0}.^{[1]} This use is improper because continuity and discontinuity of a function are concepts defined only for points in the function's domain. Such an undefined point is properly a removable singularity.
1. Consider the function
Then, the point x_{0} = 1 is a removable discontinuity.
2. Consider the function
Then, the point x_{0} = 1 is a jump discontinuity.
3. Consider the function
Then, the point x_{0} = 1 is an essential discontinuity. For it to be an essential discontinuity, it would have sufficed that only one of the two onesided limits did not exist or were infinite. However, given this example the discontinuity is also an essential discontinuity for the extension of the function into complex variables.
The set of points at which a function is continuous is always a G_{δ} set. The set of discontinuities is an F_{σ} set.
The set of discontinuities of a monotone function is at most countable. This is Froda's theorem.
Thomae's function is discontinuous at every rational point, but continuous at every irrational point.
The indicator function of the rationals, also known as the Dirichlet function, is discontinuous everywhere.
