3.1 • DIFFERENTIABLE AND ANALYTIC FUNCTIONS 95form z = xo + iy.
). f(z)-f(zo)
1
. f(xo+ iy)- f(xo+iyo)
1m = un
z-->zo z -zo (zo+iy)-(xo+iyo) (xo + iy) - (xo + iyo)
1. (xo -iy) - (xo -iyo)
Im
(zo+iY)-(xo+i110) (xo -xo) + i (y -Yo)
= Jim -i(y-yo)
(>:o+iy)-(:i;o+iyo) i (y -Yo)
= -1.The limits along the two paths are different, so there is no possible value for
the right side of Equation (3-1). Therefore, f (z) = z is not differentiable at the
point zo, and since zo was ar bitrary, f (z) is nowhere differentiable.
R e mark 3.1 In Section 2.3 we showed that f (z) = z is continuous for all z.
Thus, we have a simple example of a function that is continuous everywhere but
differentiable nowhere. Such functions are hard to construct in real variables. In
some sense, the complex case has made pathological constructions simpler! •
We seldom are interested in studying functions that aren't differentiable, or
are differentiable at only a single point. Complex functions that have a derivative
at all points in a neighborhood of zo deserve further study. In Chapter 7 we
demonstrate that if the complex function f can be represented by a Taylor series
at zo, then it must be differentiable in some neighborhood of zo. Functions that
are differentiable in neighborhoods of points are pillars of the complex analysis
edifice; we give them a special name, as indicated in the following definition.
Definition 3.1: AnalyticWe say that the complex function f is analytic at the point zo, provided there
is some e > 0 such that f' (z) exists for all z E De (z 0 ). In other words, f must
be differentiable not only at zo, but also at all points in some c neighborhood of
zo.
If f is analytic at each point in the region R , then we say tha~ f is analytic
on R. Again, we have a special term if f is analytic on the whole complex plane.
Definition 3.2: EntireIf f is analytic on the whole complex plane, then f is said to be entire.