74 Chapter 1 Fourier Series and Integrals
(a) (b) (c)Figure 7 Three functions with different kinds of discontinuities at x=1.
(a)f(x)=(x−x^2 )/( 1 −x)has a removable discontinuity. (b)f(x)=xfor
0 <x<1andf(x)=x−1for1<x; this function has a jump discontinuity.
(c)f(x)=−ln(| 1 −x|)has a “bad” discontinuity.
Theleft-hand limitis defined similarly:
f(x 0 −)=h→lim 0 −f(x 0 +h)=hlim→ 0
h< 0f(x 0 +h)=h→lim 0 +f(x 0 −h).Note thatf(x 0 +)andf(x 0 −)need not be values of the functionf.
If both left- and right-hand limits exist and are equal, the ordinary limit
exists and is equal to the one-handed limits. It is quite possible that the left-
and right-handed limits exist but are different. This happens, for instance, at
x=0 for the function
f(x)={ 1 , 0 <x<π,
− 1 , −π<x<0.In this case, the left-hand limit atx 0 =0is−1, whereas the right-hand limit
is+1. A discontinuity at which the one-handed limits exist but do not agree is
called ajump discontinuity.
It is also possible that at some point both limits exist and agree but that the
function is not defined at that point or its value is not equal to the limit. In
suchacase,afunctionissaidtohavearemovable discontinuity.Ifthevalueof
the function at the troublesome point is redefined to be equal to the limit, the
function will become continuous. For example, the functionf(x)=sin(x)/x
has a removable discontinuity atx=0. The discontinuity is eliminated by re-
definingf(x)=sin(x)/x(x=0),f( 0 )=1. Removable discontinuities are so
simple that we may assume they have been removed from any function under
discussion.
Other discontinuities are more serious. They occur if one or both of the
one-handed limits fail to exist. Each of the functions sin( 1 /x),e^1 /x,1/xhas a
discontinuity atx=0thatisneitherremovablenorajump(seeFig.7).Table2
summarizes continuity behavior at a point.