Differentiation 311
1. f is analytic on E if there is an open set A ::J E and a function f 1
analytic on A such that Ji IE = f.
2. f is locally analytic on E if for each a E E there is a neighborhood
N 0 (a) and a function fa:N 0 (a) -+ C analytic at a and such that fa I
(No(a) n E) = f I (No( a) n E). See [84].6.2 Continuity and Differentiability
Differentiability of a function at a point is a stronger condition than
continuity. In fact, we have the following theorem.
Theorem 6.1 Let f(z) be defined on an open set A. If f(z). has an
ordinary derivative at z 0 E A, then f(z) is continuous at z 0 • However, a
continuous function at z 0 may fail to have a derivative (in the ordinary
sense) at z 0 •
Proof From the identity
f(zo + h) = f(zo) + f(zo + hl-f(zo) h
we obtain, taking limits as h --t O,
lim f(zo + h) = f(zo) + f'(zo) · 0 = f(zo)
h-+Owhich shows that f(z) is continuous at zo.
It is easy to give an example of an everywhere continuous function in the
complex plane that fails to have a derivative at every point. For instance,
the function f(z) = z is obviously continuous everywhere, yet it is not
differentiable for any z since
--""-----'-----"--'-f(z + 6.z) - f(z) - z + 6.z - z - -6.z - e -zilJ
6.z - 6.z - 6.z -by letting 6.z = reilJ. Hence the limit of the left-hand side does not exist
in the usual sense because different values would be obtained by choosing
different constant values for () and letting r --t 0.
6.3 Differentiation Rules
Since Definition 6.1 of the ordinary derivative of a complex function is
formally the same as in the real case, the usual differentiation rules of
calculus apply. In fact, we have:
Theorem 6.2 Let f,g: A --t B, A open. Then: