The Extreme-Value Theorem.
THEOREM 22.1:
Let f : D →
R be continuous. If D is compact,
then f (D) is compact.
40
Def.
Let f : D → R. f (x0) is
the minimum value of f on D if
f (x0) ≤ f (x) for all x ∈ D; f (x1)
is the maximum value of f on D
if f (x) ≤ f (x1) for all x ∈ D.
41
Corollary 1.
If f : D → R is con-
tinuous and D is compact, then f
has a maximum value and a minimum value.
That is, there exist
points x0, x1 ∈ D such that
f (x0) ≤ f (x) ≤ f (x1) for all x ∈ D.
42
Corollary 2.
If
f : D → R
is
continuous and D is compact, then
f (D) is closed and bounded.
43
The Intermediate-Value Theorem.
THEOREM 22.3:
Let f : [a, b] →
R be continuous and suppose that
f (a) 6= f (b). If k is a number between f (a) and f (b) then there
is at least one point c ∈ (a, b) such
that f (c) = k.
45