Removable Discontinuity -- From Wolfram MathWorld

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A real-valued univariate function f=f(x) is said to have a removable discontinuity at a point x_0 in its domain provided that both f(x_0) and

lim_(x->x_0)f(x)=L<infty (1)

exist while f(x_0)!=L. Removable discontinuities are so named because one can "remove" this point of discontinuity by defining an almost everywhere identical function F=F(x) of the form

F(x)={f(x) for x!=x_0; L for x=x_0, (2)

which necessarily is everywhere-continuous.

RemovableDiscontinuity

The figure above shows the piecewise function

f(x)={(x^2-1)/(x-1) for x!=1; 5/2 for x=1, (3)

a function for which lim_(x->1-)f(x)=lim_(x->1+)f(x)=2 while f(1)=5/2. In particular, f has a removable discontinuity at x=1 due to the fact that defining a function F(x) as discussed above and satisfying F(1)=2 would yield an everywhere-continuous version of f.

Note that the given definition of removable discontinuity fails to apply to functions f for which lim_(x->x_0)f(x)=L and for which f(x_0) 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., f(x)=sin(x)/x has a removable discontinuity at the point x=0. 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.

Tag » What Is A Removable Discontinuity