Infinite Discontinuity

A real-valued univariate function f=f(x) is said to have an infinite discontinuity at a point x_0 in its domain provided that either (or both) of the lower or upper limits of f fails to exist as x tends to x_0.

Infinite discontinuities are sometimes referred to as essential discontinuities, phraseology indicative of the fact that such points of discontinuity are considered to be "more severe" than either removable or jump discontinuities.


The figure above shows the piecewise function

 f(x)={sin(1/x)   for x!=0; (11)/(10)   for x=0,

a function for which both lim_(x->0-)f(x) and lim_(x->0+)f(x) fail to exist. In particular, f has an infinite discontinuity at x=0.


It is not uncommon for authors to say that univariate functions f=f(x) defined on a domain D subset R and admitting vertical asymptotes of the form x=c have infinite discontinuities there though, strictly speaking, this terminology is incorrect unless such functions are defined piecewise so that c in D. For example, the function f(x)=tan(x) has vertical asymptotes at x=npi/2, n in Z, though it has no discontinuities of any kind on its domain.

Unsurprisingly, one can extend the above definition to infinite discontinuities of multivariate functions as well.

See also

Branch Cut, Continuous, Discontinuity, Discontinuous, Discontinuous Function, Essential Singularity, Isolated Singularity, Jump Discontinuity, Polar Coordinates, Pole, Removable Discontinuity, Removable Singularity, Singular Point, Singularity

This entry contributed by Christopher Stover

Explore with Wolfram|Alpha

Cite this as:

Stover, Christopher. "Infinite Discontinuity." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

Subject classifications