222 CHAPTER 12. VECTOR VALUED FUNCTIONS

12.4.1 Sufficient Conditions For ContinuityThe next theorem is a fundamental result which allows less worry about the ε δ definitionof continuity.

Theorem 12.4.2 The following assertions are valid.

1. The function af+bg is continuous at x whenever f , g are continuous at x∈D(f)∩D(g) and a,b ∈ R.

2. If f is continuous at x, f (x) ∈D(g)⊆Rp, and g is continuous at f (x) ,then g ◦fis continuous at x.

3. If f = ( f1, · · · , fq) : D(f)→ Rq, then f is continuous if and only if each fk is acontinuous real valued function.

4. The function f : Rp→ R, given by f (x) = |x| is continuous.

The proof of this theorem is in the last section of this chapter. Its conclusions are notsurprising. For example the first claim says that (af +bg)(y) is close to (af +bg)(x)when y is close to x provided the same can be said about f and g. For the second claim,if y is close to x, f (x) is close to f (y) and so by continuity of g at f (x), g (f (y)) isclose to g (f (x)). To see the third claim is likely, note that closeness in Rp is the same ascloseness in each coordinate. The fourth claim is immediate from the triangle inequality.

For functions defined on Rn, there is a notion of polynomial just as there is for functionsdefined on R.

Definition 12.4.3 Let α be an n dimensional multi-index. This means

α = (α1, · · · ,αn)

where each α i is a natural number or zero. Also, let

|α| ≡n

∑i=1|α i|

The symbol xα meansxα ≡ xα1

1 xα22 · · ·x

αn3 .

An n dimensional polynomial of degree m is a function of the form

p(x) = ∑|α|≤m

dαxα.

where the dα are real numbers.

The above theorem implies that polynomials are all continuous.