8.6 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 case.
Theorem 8.6.1 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 later in this chapter.
Theorem 8.6.2 The following assertions are valid.
- The function af + bg is continuous at x when f, g are continuous at x ∈ D
a,b ∈ ℝ.
- If and f and g are each real 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
⊆ ℝp, and g is continuous at f
, then g∘f is continuous
- If f = :
→ ℝq, then f is continuous if and only if each fk is a continuous
real valued function.
- The function f : ℝp → ℝ, given by f =