1549901369-Elements_of_Real_Analysis__Denlinger_

5.1 Continuity of a Tunction at a Point 231

(f) [_ is continuous at xo, if g(xo) #-0.

g

Proof. Exercise 14. •

Theorem 5.1.14 (Composite Functions)

(a) Suppose f is continuous at xo and g is continuous at f (x 0 ). Then the

composite function g o f is continuous at xo.

(b) Suppose lim f(x) =Yo E 'D(g), and g is continuous at Yo· Then

X-tXQ

X-tXo lim g (f(x)) = g ( Xlim -+Xo f(x)) = g(yo).

(c) Same as (b}, with x---+ xo replaced by x---+ +oo (or -oo).

V(f)

goj

Figure 5.2

Proof. (a) Suppose f is continuous at xo and g is continuous at f(xo). Let
c > 0. Since g is continuous at f(xo), 38 > 0 3 'iy E 'D(g),

IY - f(xo)I < 8 =? lg(y) - g (f(xo))I < c. (1)

Since f is continuous at xo, 3 8' > 0 3 'ix E 'D(g o !),

Ix - xol < 8! =? lf(x) - f(xo)I < 8

=? lg (f(x)) - g (f(xo))I < c by (1)

=? l(g o f)(x) - (go f)(xo)I < c.

Therefore, g o f is continuous at xo.

(b) Suppose lim f(x) =Yo E 'D(g), and g is continuous at YO· Let c > 0.
X-+Xo
Since g is continuous at Yo, 3 8 > 0 3 Vy E 'D(g), IY-Yol < 8 =? lg(y)-g(yo)I <
c. Since lim f(x) =Yo, 3 8' > 0 3 'ix E V(g o !),
X--+Xo