The first way that a function can fail to be continuous at a point a is that
lim | f(x) = L exists (and is finite) |
x --> a |
but f(a) is not defined or f(a) L. Discontinuities for which the limit of f(x) exists and is finite are called removable discontinuities for reasons explained below.
f(a) = | lim | f(x). |
x --> a |
This has the effect of removing the discontinuity.
then the discontinuity at x=a can be removed by defining f(a)=L. |
lim | g(x) | = | lim | x+1 = 2. |
x --> 1 | x --> 1 |
If the limit as x approaches a exists and is finite and f(a) is defined but not equal to this limit, then the graph has a hole with a point misplaced above or below the hole. This discontinuity can be removed by re-defining the function value f(a) to be the value of the limit.
then the discontinuity at x=a can be removed by re-defining f(a)=L. |
As and example, the piecewise function in the second equipment check on the page "Defintion of Continuity" was given by
{ | Undefined | Unless 0 < x < 1 | |
h(x) = | 3 | If x=.5 | |
1.5 + 1/(x + .25) | 0 < x < 1, x.5 |
We can remove the discontinuity by re-defining the function so as to fill the hole. In this case we re-define h(.5) = 1.5 + 1/(.75) = 17/6.
COVER | CQ DIRECTORY | HUB | CQ RESOURCES |
©
CalculusQuest^{TM}
Version 1
All rights reserved---1996
William A. Bogley
Robby Robson