082-Engineering-Mathematics-Anthony-Croft-Robert-Davison-Martin-Hargreaves-James-Flint-Edisi-5-2017

2.3 Basic concepts of functions 65
f (x)
f(t)
3
2
x
1
Figure2.11
Thefunction f (x) = 1 has a discontinuity at
x
x=0.
1 3 t
Figure2.12
Thefunction f (t)isapiecewise continuous
functionwith adiscontinuity att = 1.
If the graph ofafunction, f (x), containsabreak, then f (x) isdiscontinuous.
Afunctionwhosegraph has no breaksisacontinuousfunction.
Sometimes a function is defined by different rules on different intervals of the domain.
For example,consider
g(t)
6
f(t)=
{
2 0t<1
t 1t3
Thedomainis[0,3]buttheruleon[0,1)isdifferenttothaton[1,3].Thegraphof f (t)
is shown in Figure 2.12. Recall the convention of using • to denote that the end-point
isincludedand ◦todenotetheend-pointisexcluded.Notethat f (t)hasadiscontinuity
att = 1. Each component, or piece, of the graph is continuous and f (t) is said to be
piecewisecontinuous.
2
Apiecewisecontinuousfunctionhasafinitenumberofdiscontinuitiesinanygiven
interval.
1 3 t
Figure2.13
The functiong(t)is a
continuous function
on (0,3).
Notallfunctionsdefineddifferentlyondifferentintervalsarediscontinuous.Forexample,
{
2 0<t<1
g(t) =
2t 1t<3
isacontinuousfunction onthe interval (0,3), asshown inFigure 2.13.
2.3.9 Periodicfunctions
Aperiodicfunctionisafunctionwhichhasadefinitepatternwhichisrepeatedatregular
intervals. More formally wesay a function, f (t),isperiodicif
f(t)=f(t+T)
forall values oft. Theconstant,T, isknown asthe period ofthe function.