A real-valued univariate function is said to have a removable discontinuity at a point in its domain provided that both and
(1)
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exist while . Removable discontinuities are so named because one can "remove" this point of discontinuity by defining an almost everywhere identical function of the form
(2)
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which necessarily is everywhere-continuous.
The figure above shows the piecewise function
(3)
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a function for which while . In particular, has a removable discontinuity at due to the fact that defining a function as discussed above and satisfying would yield an everywhere-continuous version of .
Note that the given definition of removable discontinuity fails to apply to functions for which and for which fails to exist; in particular, the above definition allows one only to talk about a function being discontinuous at points for which it is defined. This definition isn't uniform, however, and as a result, some authors claim that, e.g., has a removable discontinuity at the point . This notion is related to the so-called sinc function.
Among real-valued univariate functions, removable discontinuities are considered "less severe" than either jump or infinite discontinuities.
Unsurprisingly, one can extend the above definition in such a way as to allow the description of removable discontinuities for multivariate functions as well.
Removable discontinuities are strongly related to the notion of removable singularities.