5.2 Discontinuities and Monotone Functions 237(c) Vr E Q, Vx E JR, f(rx) = [f(x)t.
( d) If f is continuous at 0, then it is continuous everywhere.
5.2 Discontinuities and Monotone Functions
Definition 5.2.1 (Continuity from the Left at a Point) Suppose f
V(f) -> JR, and x o E V(f). Then f is continuous from the left at x 0 if
Ve> 0 :lb"> 0 3 Vx E V(f), xo -8 < x < xo:::::? lf(x)-f(xo)I < e.
Definition 5.2.2 (Continuity from the Right at a Point) Suppose f :
V(f) -> JR, and xo E V(f). Then f is continuous from the right at x 0 if
Ve> 0 :lb> 0 3 Vx E V(f), xo < x < x o + 8:::::? lf(x) - f(xo)I < e.
Notes: (1) In Definitions 5.2.l and 5.2.2, xo need not be a cluster point of
V(f) but must be in V(f).
(2) In case x 0 is a cluster point of V(f) n (-oo,x 0 ), then Definition 5.2.l
is equivalent to saying that f is continuous from the left at x 0 if
f(x 0 ) = lim f(x) = f(xo).
x-+x()If x 0 is a cluster point ofV(f)n(xo, +oo), then Definition 5.2.2 is equivalent
to saying that f is continuous from the right at xo if
f(xci) = lim f(x ) = f(xo).
X-+Xt(3) f is said to have one-sided continuity at x 0 if it is either continuous
from the left at x 0 or continuous from the right at x 0.
Example 5.2.3 Consider the function f(x) = {-l ~f x:::;
2
}.
1 ifx>2
5.3.)
(See FigureThis function is continuous from the left at 2, since lim f(x ) = -1 = f(2).
x-+2-
However , this function is not continuous from the right at 2, since lim f(x) =
X->2+
1 =I-f(2). 0