1549901369-Elements_of_Real_Analysis__Denlinger_

232 Chapter 5 • Continuous Functions

0 < Ix - xol < 8' =? lf(x) - Yol < 8

=? lg (f(x)) - g (yo) I < c:.

Therefore, lim g (f(x)) = g(yo) = g ( lim f(x)).
x-~ x-~
(c) Exercise 15. •

Corollary 5.1.15 Suppose f and g are continuous at a point xo E IR. Then

(a) J1 is continuous at xo (assuming f(x) ~ 0 in some nbd. ofxo);

(b) lfl is continuous at x 0 ;

(c) max{f,g}^2 is continuous at xo;

(d) min{f,g} is continuous at xo.

Proof. Exercises 16 and 17. •

TRIGONOMETRIC FUNCTIONS

We shall not define sin x and cos x here. That is a task for later chapters.^3

After the definition is given it will be routine to prove that \:/x E JR.,

I sin xi s; lx l and lcosxl s; 1.

Also, it will be possible to establish the identity

\:/x, y E JR., sinx - sin y = 2 sin ( x; y) cos ( x; y).

Combining the above inequalities and identity, we have the identity

!sin x - sin YI = 2 lsin ( x ; y) I I cos ( x ; y) I

l

s; 2. -2-x-yl · 1

= lx-yl.

2. For definitions ofmax{f, g} and min{f,g} see Definition B.3.1 in Appendix B.


3. See Sections 7.7 and 8.8.