We noticed in Section 2.3 that the limit of a function as x approaches a can often be found
simply by calculating the value of the function at a. Functions with this property are called
continuous at a. We will see that the mathematical deﬁnition of continuity corresponds
MATH14closely with the meaning of the word continuity in everyday language. (A continuous
process is one that takes place gradually, without interruption or abrupt change.)
1 DEFINITION A function f is continuous at a number a if
The limit of a function as x approaches a can be often found by calculating the value of
the function at “a”. Functions with this property are usually called “continuouxlat a”. ▯ f▯a▯
120 |||| CHAPTER 2 LIMITS AND DERIVATIVES
DeﬁNAs illustrated in Figure 1, if f is continuous,a” if:
V EXAMPLE 2e Where are each of the following functions discontinuous?s continuous at a:
then the points ▯x, f ▯x▯▯ on the graph of f
approach the point ▯a, f ▯a▯▯ on the graph1.Sf▯2▯ is deﬁned (that is, a is in the domain of1f )
there is no gap in the curve. 2.mlim f▯x▯ exists f(a) 2 if x ▯ 0
(a) f▯x▯ ▯xla x ▯ 2 (b) f▯x▯ ▯ ▯ x
y 1 if x ▯ 0
The deﬁnition requires thry=ƒthings if f is continuo2s at a:▯a▯
xlx ▯ x ▯ 2
ƒ (c)f▯x▯ ▯e deﬁnx ▯ 2 says that f is continu(d) f▯x▯ ▯ ▯x▯x▯ approaches f▯a▯ as x approaches a.
ap•rf(a) is deﬁned (that is, a is in the domain of f)
f(a). Thus a c1ntinuous funcif x ▯ 2as the property that a small change in x produces only a
• lim (x > a) f(x) exists small change in f▯x▯. In fact, the change in f▯x▯ can be kept as small as we please by keep-
• lim (x > a) f(x) = f(a) SOLUing the change in x sufﬁciently small.
(a) Notice thatf▯2▯is not deﬁned, so f is discontinuous at 2. Later we’ll sef isy
continuous at all other numbers. other words, f is deﬁned on an open interval containing a,
except perhaps at a), we say that f is discontinuous at a (or f has a discontinuity at a) if
This implies0that a continuous funcxio(b) Heretf▯0▯ ▯ 1ris deﬁned butll change in x
produces only a small change in f(x). f is not continuous at a.
As x approaches a, Physical phenomena are usulim f▯x▯ ▯ lims. For instance, the displacement or velocity
of a vehicle varies continuously with time, as does a person’s height. But discontinuities
If FIGURE 1ned near a (f is deﬁned on does not exist. (See Example 8 in Section 2.2.) So f is discontinuous at 0.tion 2.2, where the
we say that f is discontinuous at a (or f Heaviside function is discontinuous at 0 because limat l 0▯t▯ does not exist.]
(c) Here f▯2▯ ▯ 1 is deﬁned and
Geometrically, you can think of a function that is continuous at every number in an
Example: interval as a functi2n whose graph has no break in it. The graph can be drawn without
removing your pen from the paper.lim ▯x ▯ 2▯▯x ▯ 1▯ ▯ lim ▯x ▯ 1▯ ▯ 3
xl2 xl2 x ▯ 2 xl2 x ▯ 2 xl2
y exists. But 1 Figure 2 shows the graph of a function f. At which numbers is f discontinu-
ous? Why? lim f▯x▯ ▯ f▯2▯
SOLUTION It looks as if there is a discontinuity when a ▯ 1 because the graph has a break
so fis not continuous at 2.
there. The ofﬁcial reason that f is discontinuous at 1 is that f▯1▯ is not deﬁned.
(d) TheThe graph also has a break when a ▯ 3, but the reason for the discontinuity is differ-
because limxln ▯x▯does not exist nfis an integer. (See Example 10 and Exercise 49 in
0 x Section 2.3.), f▯3▯ is deﬁned, but limx l3f▯x▯ does not exist (because the left aMd right limits
1 2 3 4 5 are different). So f is discontinuous at 3.
Figure 3 shows the graphs of the functions in Example 2. In each case the graph can’tft and
Based on the graph, we can see that when x = 1 and when x l5
be dright limits are the same). Butom the paper because a hole or break or jump occurs in
x =FIGURE 2 x = 1, we see that it is dthe graph. The kind of discontinuity illulim f▯x▯ ▯ f▯5▯s (a) and (c) is called removable
5, this is a jump discontinuity becausbecause we could remove the discontinuity by redeﬁning f at just the single number 2.
So f is discontinuous at 5. M
[The functiont▯x▯ ▯ x ▯ 1 is continuous.] The discontinuity in part (b) is called an inﬁ-
The following graphs are more examples of jump/inﬁnite/removable discontinuity.n part (d) are called jump discontinuities because
the funNow let’s see how to detect discontinuities when a function is deﬁned by a formula.
y y y y
1 1 1 1
0 1 2 x 0 x 0 1 2 x 0 1 2 3 x