5.2 Discontinuities and Monotone Functions 239What is "removable" about a "removable discontinuity"? We shall see.
Suppose f h as a r emovable discontinuity at x 0. Define a n ew function g by
defining{
f ( X) if X # Xo }
g(x) = lim f(x) if x = x o ·
X-+XQThen g is continuous at xo since lim g(x) = lim f(x) = g(x 0 ). In a word,
x-+xo x-+xo
g "removes" the discontinuity of f.
x^2 - 4
Examples 5.2.8 (a) The function f(x) = --has a removable disconti-
x - 2
nuity at 2, since lim x
2- 2
4
exists but f(2) does not. The function g(x) = x+2
x-+2 X -
"removes" the discontinuity since it is co ntinuous at 2 and agrees with f ev-
erywhere except at 2.
. sinx...
(b) The function f(x) = --h as a removable discontmmty at 0. As
x
.. sinx..
shown m calculus, bm --= 1, but f(O) does not exist. Thus, the function
{sin x if xx;~ }x
g(x) = x "removes" the discontinuity.
1 if x = 0
Definition 5.2.9 If lim - f(x) and lim + f(x) both exist but lim -f(x) #
x-+x 0 x-+x 0 x-+x 0
lim f(x), t hen we say that f h as a jump discontinuity at x 0.
x -+xci
Example 5.2.10 The function f(x) = {-l ~f x ~
2
} described in Examples
1 ifx>2
5.2.3 and 5.2.6 has a jump discontinuity at 2, since lim f(x) = -1, while
x-+2+
lim f(x) = l. [See Figure 5.3.]
x-+2-
Definition 5.2.11 A function f is said to have a simple discontinuity^4
(or a discontinuity of the first kind) at x 0 if f has either a removable
discontinuity or a jump discontinuity at xo. Any other discontinuity of f at
x 0 is called a n essential discontinuity (or a discontinuity of the second
kind).
- T o see that a "simple" discontinuity need not look especially simple, see E xercise 12.