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1.3 Convergence of Fourier Series 75

Name Criterion

Continuity f(x 0 +)=f(x 0 −)=f(x 0 )

Removable discontinuity f(x 0 +)=f(x 0 −)=f(x 0 )

Jump discontinuity f(x 0 +)=f(x 0 −)

“Bad” discontinuity f(x 0 +)orf(x 0 −)or both fail to exist

Table 2 Types of continuity behavior atx 0

Figure 8 Typical sectionally continuous function made up of four continuous
“sections.”

We shall say that a function issectionally continuous(also calledpiecewise
continuous)onanintervala<x<bif it is bounded and continuous, ex-
cept possibly for a finite number of jumps and removable discontinuities. (See
Fig. 8.) A function is sectionally continuous (without qualification) if it is sec-
tionally continuous on every interval of finite length. For instance, if a periodic
function is sectionally continuous on any interval whose length is one period
or more, then it is sectionally continuous.

Examples.

1.Thesquare wave,definedby

f(x)=

{ 1 , 0 <x<a,

− 1 , −a<x<0, f(x+^2 a)=f(x),

is sectionally continuous. There are jump discontinuities atx=0,±a,

± 2 a,etc.

2.The functionf(x)= 1 /xcannot be sectionally continuous on any interval

that contains 0 or even has 0 as an endpoint, because the function is not

bounded atx=0.

3.Iff(x)=x,− 1 <x<1, thenfis continuous on that interval. Its periodic

extension (see Fig. 3) issectionallycontinuous but not continuous.