Calculus: Analytic Geometry and Calculus, with Vectors

3.4 Continuity 145

tion of limit. With a small change in notation, we can see that f is
continuous at x if and only if

(3.421)

or

(3.422)

lim f(x + Ox) = f(x)

AZ-.o

lim [f(x + Ax) - f(x)1 = 0.

Figure 3.43 shows, for the case in which f(x) = x2 and Ox > 0, the

geometric interpretations that can be given to the numbers appearing in

these formulas.

Figure 3.43

Definition 3.44 .4 function f is said to have right-hand continuity at a

if the first of the assertions

(3.441) lim f (x) = f (a), lim Ax) = f (b)

m-.a+ x-.b -

is valid and to have left-hand continuity at b if the second is valid.

Supposing that a < b, we can let fl be the function having the graph in

Figure 3.442 so that fi(x) = 0 when x < a, f,(x) = 1 when a < x< b,

and fi(x) = 0 when x > b. This function is continuous at each x for

which x 0 a and x 0 b. The function has right-hand continuity at a

.I----f----' -- ----f----

a b x a b x

Figure 3.442 Figure 3.443

and has left-hand continuity at b. It does not have left-hand con-

tinuity at a, and it does not have right-hand continuity at b. Let f2

be the function having the graph in Figure 3.443 so that f2(x) = 0 when

x <- a, f2(x) = 1 when a < x < b, and f2(x) = 0 when x > b. This