
“real: chapter_05” — 2011/5/22 — 22:55 — page 41 — #41
Limits and Continuity 5-41
Proof Follows from Theorems 5.8.17 and 4.3.37.
Theorem 5.8.19 Suppose f is a continuous real function on a compact
metric space X and
M = sup{f (p)/ p ∈ X }, m = inf{f (p)/ p ∈ X }.
Then there exist points p, q ∈ X such that f (p) = M and f (q) = m.
Proof By Theorem 5.8.18, f (X ) is a closed and bounded set of real
numbers. Hence f (X ) contains
M = sup f (X ), m = inf f (X )
by Theorem 4.2.18.
Theorem 5.8.20 Suppose f is a continuous one-to-one mapping of a
compact metric space X onto a metric space Y , then its inverse mapping
g : Y → X is continuous.
Proof Applying