94
5. CONTINUOUS FUNCTIONS
Theorem (5.3.7 — Bolzano’s Intermediate Value Theorem). Let I be an
interval and f : I ! R be continuous on I. If a, b 2 I and k 2 R 3
f (a) < k < f (b), then 9 c 2 I between a and b 3 f (c) = k.
Proof. WLOG, assume a < b (the other case is similar).
Let S = {x 2 [a, b] : f (x) < k}.
Since a 2 S, S 6= ;. By the Supremum Theorem,
9 c 2 [a, b] 3 c = sup S. Then
1
< xn  c.
n
Thus lim(xn) = c by the Squeeze Theorem, and since f (xn) < k 8 n 2 N,
lim f (xn)  k. Since f is continuous at c,
8 n 2 N, 9xn 2 S 3 c
f (c) = lim f (xn)  k.
[To show f (c)
k also =) f (c) = k.]
n
1
Now let yn = min b, c +
8 n 2 N.
n
1
Since c < yn  c + , lim(yn) = c by the Squeeze Theorem. [Where is yn?]
n
Now, 8 n 2 N, yn 2 [a, b]\S, so f (yn) k.
Again, by continuity at c,
f (c) = lim f (yn)
k =) f (c) = k.