Calculus 2nd Edition Rogawski

SECTION 2.4 Limits and Continuity 63
If lim x→c
f (x) exists but is not equal to f (c), we say that f has a removable discontinuity
at x = c. The function in Figure 5(A) has a removable discontinuity at c = 2
because
f(2) = 10 but lim
x→2
f (x) = 5
} {{ }
Limit exists but is not equal to function value
Removable discontinuities are “mild” in the following sense: We can make f continuous
at x = c by redefining f (c). In Figure 5(B), f(2) has been redefined as f(2) = 5,
and this makes f continuous at x = 2.
10
y
10
y
5
5
FIGURE 5 Removable discontinuity: The
discontinuity can be removed by redefining
f(2).
x
2
2
(A) Removable discontinuity at x = 2 (B) Function redefined at x = 2
x
A“worse” type of discontinuity is a jump discontinuity, which occurs if the one-sided
limits lim f (x) and lim f (x) exist but are not equal. Figure 6 shows two functions
x→c−
x→c+
with jump discontinuities at c = 2. Unlike the removable case, we cannot make f (x)
continuous by redefining f (c).
y
y
FIGURE 6 Jump discontinuities.
2
(A) Left-continuous at x = 2
x
2
(B) Neither left- nor right-continuous at x = 2
x
In connection with jump discontinuities, it is convenient to define one-sided
continuity.
DEFINITION One-Sided Continuity
• Left-continuous at x = c if lim
A function f (x) is called:
x→c−
f (x) = f (c)
• Right-continuous at x = c if lim f (x) = f (c)
x→c+
In Figure 6 above, the function in (A) is left-continuous but the function in (B) is
neither left- nor right-continuous. The next example explores one-sided continuity using a
piecewise-defined function—that is, a function defined by different formulas on different
intervals.
EXAMPLE 2 Piecewise-Defined Function Discuss the continuity of
⎧
⎪⎨ x for x3