316 CHAPTER 15. SEQUENCES, COMPACTNESS, CONTINUITY

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

Theorem 15.7.1 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.

5. The map πk (x)≡ xk is continuous.

6. Every function x→ xα11 xα2

2 · · ·xα pp for α i an integer is continuous.

7. If f,g are each continuous at x, then f ·g is also continuous at x.

This is proved just like the corresponding theorem for functions of a single variable. Forexample the first claim says that (af +bg)(y) is close to (af +bg)(x) when y is close tox 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)) is close to g (f (x)).To see the third claim is likely, note that closeness in Rp is the same as closeness in eachcoordinate. The fourth claim is immediate from the triangle inequality. Alternatively, useProposition 15.6.3 to reduce to notions of convergent sequences and then Definition 15.6.2to reduce completely to one variable considerations and apply earlier theorems on limitsand continuity.

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

Definition 15.7.2 Let α be an p dimensional multi-index. This means

α = (α1, · · · ,α p)

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

|α| ≡p

∑i=1

|α i|

The symbol xα meansxα ≡ xα1

1 xα22 · · ·xα p

3 .

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

p(x) = ∑|α|≤m

dαxα.

where the dα are real numbers.