1549901369-Elements_of_Real_Analysis__Denlinger_

238 Chapter 5 • Continuous Functions

y

2

-1

-2

2 3 4

Figure 5.3

x

Theorem 5.2.4 (Sequential Criterion for One-Sided Continuity)

(a) A function f: V(f) __,JR, is continuous from the left at a point Xo E V(f)

iff \f sequences { xn} in V(f) n (-oo, xo) 3 Xn __, xo, f (xn) __, f (xo).

(b) A function f : V(f) __, JR, is continuous from the right at a point Xo E

V(f) iff \f sequences {xn} in V(f)n(xo, +oo) 3 Xn __, xo, f(xn) __, f(xo).

Proof. Exercise 1. (Compare with Theorem 5.1.3) •

Theorem 5.2.5 Suppose f : V(f) __, JR, and xo E V(f). Then f is continu-

ous at xo iff f is continuous from the left at xo and continuous from the right

at xo.

Proof. Exercise 2. •

Example 5.2.6 The function f(x) = {-l ~f x::;

2

} described in Example

1 if x > 2

5.2.3 is not continuous at 2, since it is not continuous from the right at 2.

SOME TYPES OF DISCONTINUITIES

Definition 5.2.7 If lim f(x) exists but either lim f(x) =/= f(x 0 ) or f(x 0 )

X---tXo X---tXo

does not exist, then we say that f has a removable discontinuity at x 0.