76 Chapter 1 Fourier Series and Integrals
The examples clarify a couple of facts about the meaning of sectional con-
tinuity. Most important is that a sectionally continuous function must not
“blow up” at any point — even an endpoint — of an interval. Note also that a
function need not be defined at every point in order to qualify as sectionally
continuous. No value was given for the square-wave function atx=0,±a,
but the function remains sectionally continuous, no matter what values are
assigned for these points.
Afunctionissectionally smooth(also,piecewise smooth)inanintervala<
x<bif:f is sectionally continuous;f′(x)exists, except perhaps at a finite
number of points; andf′(x)is sectionally continuous. The graph of a section-
ally smooth function then has a finite number of removable discontinuities,
jumps, and corners. (The derivative will not exist at these points.)Between
these points, the graph will be continuous, with a continuous derivative. No
vertical tangents are allowed, for these indicate that the derivative is infinite.
Examples.
1.f(x)=|x|^1 /^2 is continuous but not sectionally smooth in any interval that
contains 0, because|f′(x)|→∞asx→0.
2.Thesquarewaveissectionallysmoothbutnotcontinuous.
Most of the functions useful in mathematical modeling are sectionally
smooth. Fortunately we can also give a positive statement about the Fourier
series of such functions.
Theorem. If f(x)is sectionally smooth and periodic with period 2 a, then at each
point x the Fourier series corresponding to f converges, and its sum is
a 0 +∑∞
n= 1ancos(
nπx
a)
+bnsin(
nπx
a)
=f(x+)+ 2 f(x−). See an animated example on the CD.
This theorem gives an answer to the question at the beginning of the section.
Recall that a sectionally smooth function has only a finite number of jumps
and no bad discontinuities in every finite interval. Hence,
f(x−)=f(x+)=^1
2(
f(x+)+f(x−))
=f(x),exceptperhapsatafinitenumberofpointsonanyfiniteinterval.Forthisrea-
son, iffsatisfies the hypotheses of the theorem, we writefequalto its Fourier
series, even though the equality may fail at jumps.
In constructing the periodic extension of a function, we never defined the
values off(x)at the endpoints. Since the Fourier coefficients are given by in-
tegrals, the value assigned tof(x)at one point cannot influence them; in that