1550251515-Classical_Complex_Analysis__Gonzalez_

Fm1ctions. Limits and Continuity. Arcs and Curves 143

----..

a

/

'

0

/ z

I / ' \

I a I

I

\ D I

\ I

'

.... __ / /

(a) (b) (c)

Fig. 3.9

in Fig. 3.9b, we may refer to continuity off at a from the interior, and if

Dis an arc, as in Fig. 3.9c, to continuity off at a along the arc D. This is

redundant, in view of Definition 3.1, but it is often done just for emphasis.

Taking into account Definition 3.1, the definition of continuity of a func-

tion at a point can be put in the following equivalent form (called the e-8

definition of continuity):

Definition 3.9 Let f: D --r <C, and let a E D be an accumulation point

of D. Then f is continuous at a iff for every e > 0 there is a 8 > 0 such that

lf(z) - f(a)I < e whenever z E Ns(a) n D (3.11-1)

There is no need now to consider a deleted neighborhood of a, since f(a)

is defined. ·

A slightly more general definition of continuity at a point is obtained

by dropping the requirement that a be· an accumulation point of D and

using (3.11-1) as the definition of continuity at a. That makes f continuous

at isolated points of D (if any), since then the condition (3.11-1) is obviously

satisfied for 8 small enough, i.e., such that Ns(a) n D ={a}. This notion

of continuity will seldom be used in what follows.
The concepts of limit and of continuity of a function at a point can
easily be extended to the case of functions defined from a metric space
(S,d) into another metric space (S',d'). For that it suffices to replace the
neighborhoods Ns(a), N,(L) [or N,(f(a)), for continuity], which are taken-
·above in the Euclidean sense, by neighborhoods in the sense of the metric
of the corresponding spaces. In particular, we may consider continuity
of a complex function with the domain space, the image space, or both,

replaced by (<C*, x), i.e., by the Riemann sphere with the chordal metric. It

should be pointed out that a function that fails to be continuous at a point

of (<C, d) (d denoting the Euclidean metric) may very well be continuous

at the same point if the image space is replaced by (<C*, x) and a suitable

extension of the definition of f is made.