1549901369-Elements_of_Real_Analysis__Denlinger_

5.1 Continuity of a Function at a Point 233

Thus (see Exercise 18) we conclude that

The sine function is continuous everywhere.

Similarly, after sin x and cos x are defined it will be possible to prove that
Vx E IR, lsinxl ::; 1, and that

Vx, y E IR, cos x -cosy= 2 sin ( x; y) sin ( y; x).

Reasoning as we did above, we use these identities to show that

l

lcosx - cosyl::; 2 · 1 · -x-yl

2

• =Ix -yl, and thus (see Exercise 18),

The cosine function is continuous everywhere.

Now the remaining trigonometric functions are defined by

sinx

tanx= --;

cosx

cosx

cotx = -.-;

smx

1

secx = --;

cosx

1

cscx = -.-.

smx

By the "algebra of continuous functions" theorem, these functions are con-
tinuous wherever they exist. The following theorem summarizes these results.

Theorem 5.1.16 The six trigonometric functions are continuous everywhere
on their domains.

Proof. For tanx, cotx, secx, and cscx, see Exercise 20. •

EXERCISE SET 5.1

1. Prove that if x 0 is an isolated point of D(J), then f is continuous at xo
   but lim J(x) does not exist. What does this tell you about the function
   X-tXo
   f(x) = .J~x3,,--_-x""""2?


2. Prove that if x 0 is a cluster point of D(J), then f is continuous at xo iff

{

(a) f(xo) exists;

(b) lim j(x) exists; and

X--+Xo

(c) lim f(x) = f(xo).

X--+Xo