2.5. COMPACTNESS AND CONTINUOUS FUNCTIONS 51
x̂∈B(x,δ ) , then ρ ( f (x̂) , f (x))< ε meaning that f (x̂)∈B( f (x) ,ε). Thus f is continuousat x for every x. ■
In the case where f : D(f) ⊆ Rp → Rq, the above definition takes the following morefamiliar form.
Definition 2.5.20 A function f : D(f) ⊆ Rp → Rq is continuous at x ∈ D(f) if foreach ε > 0 there exists δ > 0 such that whenever y ∈ D(f) and
|y−x|< δ
it follows that|f(x)− f(y)|< ε.
f is continuous if it is continuous at every point of D(f).
This is equivalent to the same statement with ∥·∥∞
in place of |·| because
∥x∥∞≤ |x| ≡
(p
∑k=1|xk|2
)1/2
≤√p∥x∥∞
and it will be shown a little later that any two norms satisfy an inequality of the above sortso the choice of norm does not affect whether a function is continuous in the sense that ifit is continuous with respect to one norm, then it is continuous for the other.
Corollary 2.5.21 f : Rp→ Rq is continuous if and only if f−1 (V ) is open in Rp when-ever V is open in Rq and f−1 (C) is closed in Rp whenever C is closed in Rq.
Recall how the function x→ dist(x,S) was continuous. Theorem 2.5.19 implies{x : dist(x,S)>
1k
}is open,
{x : dist(x,S)≥ 1
k
}is closed
and so forth.Now here are some basic properties of continuous functions which have values in Rp
or R so that it makes sense to add and multiply by scalars. However, no context is specifiedfor property 3. which holds for f,g having values and domains in metric space.
Theorem 2.5.22 The following assertions are valid.
1. The function af+bg is continuous at x when f, g are continuous at x ∈ D(f)∩D(g)and a,b ∈ R.
2. If and f and g are each real valued functions continuous at x, then f g is continuousat x. If, in addition to this, g(x) ̸= 0, then f/g is continuous at x.
3. If f is continuous at x, f(x) ∈ D(g), and g is continuous at f(x) , then g◦ f is contin-uous at x.
4. If f = ( f1, · · · , fq) : D(f)→Rq, then f is continuous if and only if each fk is a contin-uous real valued function.
5. The function f : Rp→ R, given by f (x) = |x| is continuous.