On-line Math 21

On-line Math 21

Classification of discontinuities

Discontinuities of a function f (points x where f is not continuous) can be separated into several distinct categories:

Removable discontinuities.   These are places x0 where f just has the wrong value, and that if you change the definition of f at x0 , the function would be continuous.

Jump-discontinuities.   Are points x0 so that the left- and right-hand limits exist, but are unequal. The value of the function could be either one-sided limit.

Poles.   Also called infinite discontinuities. These are places where the limits (one side or both) are infinite.

Essential singularities.   These are places where not even the one-sided limits exist, not even as infinities. Two examples would be:
f(x): = ì
í
î
1,
if x is rational
0,
if x is irrational
 ,
which is not continuous at any point, and has no one-sided limits, so has essential singularities at all points, and
g(x) = sin(1/x),
which has an essential singularity at x = 0 .

Copyright (c) 2000 by David L. Johnson.

File translated from TEX by TTH, version 2.61.
On 18 Oct 2000, 00:45.