ABSTRACT
An examination of the definition of differentiability shows that for the deri-
vative f ?(x ) of a function f (x ) to exist at a point x/x0, it is necessary that f(x0) is defined and such that
f (x)f (x0):
This establishes the important result that differentiability at a point implies
continuity at that point. The converse is untrue because continuity at x/x0 does not imply differ-
entiability at that point. A function which is continuous at a point may, or
may not, be differentiable there. This is easily seen by considering the function
f (x) ½x½, for BxB:
This function, which is shown in Fig. 66, is continuous for all x and differ-
entiable for xB/0 with f ?(x )//1, and for x/0 with f ?(x )/1, but f ?(x ) is not defined at the origin, despite the fact that f(x) is continuous there.
