5.8 Properties Of Continuous Functions
Functions of p variables have many of the same properties as functions of one variable.
First there is a version of the extreme value theorem generalizing the one dimensional
Theorem 5.8.1 Let C be closed and bounded and let f : C → ℝ be continuous. Then f
achieves its maximum and its minimum on C. This means there exist, x1,x2 ∈ C such
that for all x ∈ C,
There is also the long technical theorem about sums and products of continuous
functions. These theorems are proved in the next section.
Theorem 5.8.2 The following assertions are valid
- The function, af + bg is continuous at x when f, g are continuous at x
and a,b ∈ F.
- If and f and g are each F valued functions continuous at x, then fg is
continuous at x. If, in addition to this, g
≠0, then f∕g is continuous at x.
- If f is continuous at x, f
⊆ Fp, and g is continuous at f
g ∘ f is continuous at x.
- If f = :
→ Fq, then f is continuous if and only if each fk is
a continuous F valued function.
- The function f : Fp → F, given by f =