2.3 • LIMITS AND CONTI NUITY 75Definition 2.5 : Continuity of u (x, y)Let u (x , y) be a real-valued function of the two real variables x and y. We say
that u is continuous at the point ( xo, Yo) if three conditions are satisfied:
Lim u (x, y) exists;
(x,y)-(:to ,yo)
(2-21)
u (xo, Yo) exists; and (2-22)
Lim u(x, y) = u (:i>o, Yo).
(x,y )-(xo ,yo)
(2-23)
Condition (2-23) actually implies Conditions (2- 21) and (2-22) because the
existence of the quantity on each side of Equation (2-23) is implicitly understood
to exist. For example, if u (x, y) = ".,x;. 11 ., when (x, y) ,P (O, 0) and if u (0, 0) = 0,
then u (x , y) , 0 as (x, y) , (O, O) so that Conditions (2-21 ), (2- 22), and (2-23)
are satisfied. Hence u (x, y) is continuous at (O, 0).
There is a similar definition for complex-valued functions.
Definition 2.6: Continuity of f (z)Let f (z) be a complex function of the complex variable z that is defined for all
values of z in some neighborhood of zo. We say that f is continuous at z 0 if
three conditions are satisfied:
lim f ( z) exists;
z-zo
(2-24)
f (zo) exists; (2-25)
lim f(z)=f(zo).
z-zo (2- 26)Remark 2.3 Example 2.16 shows that the function f (z) = z is continuous. •
A complex function f is continuous iff its real and imaginary parts, u and
v, are continuous. The proof of this fact is an immediate consequence of The-
orem 2.1. Continuity of complex functions is formally the same as that of real
functions, and sums, differences, and products of continuous functions are con-
tinuous; their quotient is continuous at points where the d enominator is not zero.
These results a.re summarized by the following theorems. We leave the proofs as
exercises.