1549901369-Elements_of_Real_Analysis__Denlinger_

5.3 Continuity on Compact Sets and Intervals 251

For all n E N, we define tn = min { c + ~, b}. Thus, 'Vn E N, c < tn :::; c + ~.

Hence, by the squeeze principle, tn _.., c. (5)

(tll

a c b

t

=sup A

Figure 5.8

Now f is continuous at c and Vn, tn E [a, b]. Thus, by (5) and the sequential
criterion for continuity of f at c,

f (tn) _.., f ( c). (6)

But, 'Vn, tn > c = sup A, so tn tj. A, which by definition of A means that

f (tn) 2: w. Since limits preserve inequalities, lim n-HXl f (tn) 2: w. That is , by (6),

I f(c) 2: w. I (7)

Putting (4) and (7) together, we have f(c) = w. That is, w E f(I).

Therefore, f (I) is an interval.

Case 2 (b <a): The proof of this case is just like that of Case 1, with the
roles of a and b interchanged. •

Corollary 5.3.9 (Intermediate Value Theorem) Suppose a < b. Any con-
tinuous f : [a, b] _..,JR must satisfy the intermediate value property on [a, b]:

Vy between f(a) and f(b), 3 c E [a, b] 3 f(c) = y.

y

x

Figure 5.9