CHAPTER 7
Continuity
7.2
Find the points of discontinuity (if any) of the function f(x) whose graph is shown in Fig. 7-1.
x = 0 is a point of discontinuity because lim f(x) does not exist, x = 1 is a point of discontinuity
because lim f(x)*f(l)
[since lim/(jt) = 0 and /(I) = 2].
7.3
Determine the points of discontinuity (if any) of the function f(x) such that f(x) = x
2
if x =£ 0 and f(x) - x
if x>0.
f(x) is continuous everywhere. In particular, f(x) is continuous at x = 0 because /(O) = (O)
2 = 0 and
lim f(x) = 0.
*-»0
7.4
Determine the points of discontinuity (if any) of the function/(*) such that f(x) = 1 if x^O and /(jt)=-l
if x<0. (See Fig. 7-2.)
Fig. 7-2
/(*) is not continuous at x = 0 because lim f(x) does not exist.
7.5
Determine the points of discontinuity (if any) of the function f(x) such that f(x) =
fix) = 0 if x=-2. (See Fig. 7-3.)
Since x
2 -4 = (x -2)(x + 2), f(x) = x-2 if x *-2. So, /(*) is not continuous at x=-2 because
lim_^f(x)*f(-2) [since /(-2) = 0 but jmi2/(A:) =-4]. [However, j: =-2 is called a removable discontinuity, because, if we redefine f(x) at x= -2 by setting /(-2) = -4, then the new function is
continuous at x = -2. Compare Problem 7.2.]
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7.1
Define: f(x) is continuous at x - a.
f(a) is defined,
exists, and
Fig. 7-1
and
if