# Discontinuity

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## discontinuity

[dis‚känt·ən′ü·əd·ē]## Discontinuity

(or point of discontinuity), a value of the argument at which the continuity of a function is violated. In the simplest case, continuity is violated at some point *a* in the sense that the right and left limits

exist, but at least one of them differs from *f(a).* When this occurs, *a* is called a jump discontinuity of *f,* or the discontinuity of the first kind.

If *f*(*a* + 0) = *f(a* – 0), the discontinuity is said to be removable, since *f(x)* becomes continuous at *a* if we set *f(a)* = *f*(*a* + 0) = *f*(*a* – 0). For example, the point *a* = 0 is a removable discontinuity of the function

since *f* is continuous at 0, if we set *f*(0) = 1. If, however, the jump *δ* = *f*(*a* + 0) – *f(a* – 0) of the function *f*(*x*) at the point is nonzero, then *a* is a discontinuity for any definition of the value of *f(a).* An example of such a discontinuity is the point *a* = 0 for the function *f(x)*= arc tan 1/*x.* In this case, the function may not be defined at the point *a.* The jump discontinuity is called regular if the condition *f(a)* = ½[*f*(*a* – 0) + *f*(*a* + 0)] is satisfied. If either of the one-side limits does not exist, then the point *a* is called the discontinuity point of the second kind. Examples are the point *a* = 2 for the function *f(x)* = 1/(*x* – 2) and the point *a* = 0 for f(x) = sin 1/*x.*