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44. Suppose that and , where is
a real number. Prove each statement.
(a) limxla
fx t x
climxlat x climxl afx (b) if
(c) if c 0limxla
fx t x
c 0limxla
fx t x
|||| 2.5 Con t i nu i t y
We noticed in Section 2.3 that the limit of a function as
approaches can often be found
simply by calculating the value of the function at . Functions
with this property are called
continuous at a. We will see that the mathematical definition of
continuity corresponds
closely with the meaning of the word continuity in everyday
language. (A continuous
process is one that takes place gradually, without interruption
or abrupt change.)
Definition A function is continuous at a number a if
Notice that Definition l implicitly requires three things if is
continuous at a:
1. is defined (that is, a is in the domain of )
2. exists
3.
The definition says that is continuous at if approaches
asxapproaches a.
Thus, a continuous function has the property that a small change
inxproduces only a
small change in . In fact, the change in can be kept as small as
we please by keep-
ing the change in sufficiently small.
If is defined near (in other words, is defined on an open
interval containing ,
except perhaps at ), we say that is discontinuous at a, or has a
discontinuity at , if
is not continuous at .
Physical phenomena are usually continuous. For instance, the
displacement or velocity
of a vehicle varies continuously with time, as does a persons
height. But discontinuities
do occur in such situations as electric currents. [See Example 6
in Section 2.2, where the
Heaviside function is discontinuous at because does not
exist.]
Geometrically, you can think of a function that is continuous at
every number in an
interval as a function whose graph has no break in it. The graph
can be drawn without
removing your pen from the paper.
EXAMPLE 1 Figure 2 shows the graph of a function f. At which
numbers isfdiscontinu-
ous? Why?
SOLUTION It looks as if there is a discontinuity when a 1
because the graph has a break
there. The official reason thatfis discontinuous at 1 is that is
not defined.The graph also has a break when , but the reason for
the discontinuity is differ-
ent. Here, is defined, but does not exist (because the left and
right limits
are different). Sofis discontinuous at 3.
What about ? Here, is defined and exists (because the left
and
right limits are the same). But
So is discontinuous at 5.f
limxl5
fx f 5
limxl5fxf 5a 5
limxl3fxf 3
a 3f 1
lim tl 0H t0
af
affa
afaf
x
fxfxf
f afxaf
limxl
afx f a
limxl
afx
ff a
f
limxl
afx f a
f1
a
ax
|||| As illustrated in Figure 1, if is continuous,
then the points on the graph of
approach the point on the graph. So
there is no gap in the curve.
a,f a
f x,fx
f
f(a)
x0
y
a
y=
approaches
f(a).
As xapproaches a,
FIGURE 1
Explore continuous functions interactively.
Resources / Module 2
/ Continuity
/ Start of Continuity
FIGURE 2
y
0 x1 2 3 4 5
