Page 34 - Calculus with Complex Numbers

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Similarly one can inprinciple go through all the elemental'y functions of calculus
and show they have the derivatives they are supposed to have.
W e can also prove all the elemental'y combination rules for differentiating sums,
products quotients and composites.
We cannot assume that the derivative f' @) always exists. For example, if
f @) = 1.1 I then

so has no limit as h --> 0.
Observe that the graph of f @) = 1.x I has no well defined tangent at .x = 0 (see
Figure 3.2).
We therefore deline f @) to be d@erentiable at .x if

li f (x + h) - f (x )
m
à..+.0 h
exists. According to this delinition flx) = 1.x I is not differentiable at .x = 0.
Another case where differentiability fails is at a discontinuity of flx) . A
continuous function flx) is one whose graph has no breaks. We make this idea
precise by delining flx) to be continuous at .x if

lim f (x + h) = f @) .
à--+0

For example, f @) = 1/.x is not continuous at .x = 0 (Figure 3.3). ln this
connection we have the following theorem.

Theorem 1 Differentiability implies continuity.