13.6. CONVERGENCE OF FUNCTIONS 243
and g would be a continuous and unbounded function defined on C, contrary to Lemma13.5.2. Therefore, there exists x2 ∈C such that f (x2) = M. A similar argument applies toshow the existence of x1 ∈C such that
f (x1) = inf{ f (x) : x ∈C} . ■
As in the case of a function of one variable, there is a concept of uniform continuity.
Definition 13.5.4 A function f : D(f)→ Rq is uniformly continuous if for every ε > 0there exists δ > 0 such that whenever x,y are points of D(f) such that |x−y| < δ , itfollows |f (x)−f (y)|< ε .
Theorem 13.5.5 Let f : K → Rq be continuous at every point of K where K is a closedand bounded set in Rp. Then f is uniformly continuous.
Proof: Suppose not. Then there exists ε > 0 and sequences{x j}
and{y j}
of pointsin K such that ∣∣x j−y j
∣∣< 1j
but∣∣f (x j)−f
(y j)∣∣≥ ε . Then by Corollary 13.3.9 on Page 241 which says K is sequen-
tially compact, there is a subsequence{xnk
}of{x j}
which converges to a point x ∈ K.Then since
∣∣xnk −ynk
∣∣< 1k , it follows that
{ynk
}also converges to x. Therefore,
ε ≤ limk→∞
∣∣f (xnk
)−f
(ynk
)∣∣= |f (x)−f (x)|= 0,
a contradiction. Therefore, f is uniformly continuous as claimed. ■
13.6 Convergence of FunctionsThere are two kinds of convergence for a sequence of functions described in the next defi-nition, pointwise convergence and uniform convergence. Of the two, uniform convergenceis far better and tends to be the kind of convergence most encountered in complex analy-sis. Pointwise convergence is more often encounted in real analysis and necessitates muchmore difficult theorems.
Definition 13.6.1 Let S ⊆ Cp and let fn : S→ Cq for n = 1,2, · · · . Then {fn} is said toconverge pointwise to f on S if for all x ∈ S,
fn (x)→ f (x)
for each x. The sequence is said to converge uniformly to f on S if
limn→∞
(supx∈S|fn (x)−f (x)|
)= 0
supx∈S |fn (x)−f (x)| is denoted as∥fn−f∥∞
or just ∥fn−f∥ for short.
∥·∥
is called the uniform norm.