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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.

Contents

Classification of discontinuities

Consider a function ƒ of real variable x with real values defined in a neighborhood of a point x0. Then three situations are possible in which the function ƒ is discontinuous at the point on the real axis x = x0:

  1. The one-sided limit from the negative direction
    L^{-}=\lim_{x\rarr x_0^{-}} f(x)
    and the one-sided limit from the positive direction
    L^{+}=\lim_{x\rarr x_0^{+}} f(x)
    at x0 exist, are finite, and are equal. Then, if f(x0) is not equal to L = L + , x0 is called a removable discontinuity. This discontinuity can be 'removed to make f continuous at x0', or more precisely, the function
    g(x) = \begin{cases}f(x) & x\ne x_0 \\ L & x = x_0\end{cases}
    is continuous at x=x0.
  2. The limits L and L + exist and are finite, but not equal. Then, x0 is called a jump discontinuity or step discontinuity. For this type of discontinuity, the value of f(x0) does not matter.
  3. One or both of the limits L and L + does not exist or is infinite. Then, x0 is called an essential discontinuity, or infinite discontinuity. (This is distinct from the use of the term when applied to a function of a complex variable.)

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 x0.[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.

Examples

The function in example 1, a removable discontinuity

1. Consider the function

f(x)=\begin{cases}x^2 & \mbox{ for } x< 1 \\ 0 & \mbox { for } x=1 \\ 2-x& \mbox{ for } x>1\end{cases}

Then, the point x0 = 1 is a removable discontinuity.

The function in example 2, a jump discontinuity

2. Consider the function

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 x0 = 1 is a jump discontinuity.

The function in example 3, an essential discontinuity

3. Consider the function

f(x)=\begin{cases}\sin\frac{5}{x-1} & \mbox{ for } x< 1 \\ 0 & \mbox { for } x=1 \\ \frac{0.1}{x-1}& \mbox{ for } x>1\end{cases}

Then, the point x0 = 1 is an essential discontinuity. For it to be an essential discontinuity, it would have sufficed that only one of the two one-sided 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 discontinuities of a function

The set of points at which a function is continuous is always a Gδ set. The set of discontinuities is an Fσ 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.

See also

Notes

  1. ^ See, for example, the last sentence in the definition given at Mathwords.[1]

References

  • Malik, S. C.; Arora, Savita (1992). Mathematical analysis, 2nd ed. New York: Wiley. ISBN 0470218584.  

External links

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