Understanding Engineering Mathematics

A functionf(x)such that

lim

x→a

f(x)

=f(a)

is said to bediscontinuous atx=a.

For example,f(x)=

x+ 3

(x− 1 )(x+ 5 )

is discontinuous atx=1andx=−5. Another

example of a discontinuous function is the tanxfunction (see Figure 6.10).
In general, discontinuous functions are a nuisance in mathematics, but various tech-
niques have been devised to deal with them. They are sometimes useful to approxi-
mate rapid continuous variations, as for example the use of thestep function,H(t)= 0
fort<0and=1fort>0, in representing the ‘instantaneous’ throwing of a switch
(Figure 14.6).

H(t)

0 t

Figure 14.6The step functionH(t).

Although we have used limits to define continuity, in practice we often know that a
given functionf(x)is continuous, and then we can simply putx=ato find the ‘limit’
or value off(x)atx=a. For example, the general rational functionP(x)/Q(x)can
only have discontinuities whereQ(x)=0 and these are infinite ifP(x)is non-zero. If
however, atx=a,P(a)=Q(a)=0, then we have an indeterminate form 0/0 (the same
discussion applies for∞/∞) and a more detailed study is required. This can take the form
of a substitution such asz=a+hand studying the limit ash→0. This is equivalent
to focusing attention on the pointx=aand looking closely at the behaviour near this
point. We can perform any algebraic operations onP(x)/Q(x)to simplify it, or rearrange
it to a more convenient form for taking the limit – we can for example, cancel factors like
x−a, because we never actually consider what happens atx=a. Another approach is to
expand the function in a power series. In fact our main concern with limits and continuity
lies in the theory of differentiation, where we need to consider indeterminate expressions
such as 0/0.

Problem 14.6

Investigate the continuity of the function

f.x/=

x− 3

x^2 − 4 xY 3

Sketch the function and define a new function,g.x/, that is continuous

and such thatg.x/=f.x/for allx>2, except atx=3.