300 Chapter 6 • Differentiable Functions... f(x) - f(xo)
(See Exercise 4.1.6.) Because of this, if we let g(x) = , then we
x -xo
can rewrite Definition 6.1.l in the following alternate form:
Definition 6.1.6 (Alternate Definition of Differentiability) Suppose
f: D(f) --t IR and xo is an interior point of D(f). Then f is differentiable at
xo ·f i t e h 1. im1t · 1. im f (xo + h) h - f (xo) exists · (. i.e., · is fi mte · ). If t h. is 1. im1 ·t ex1s · t s,
h-+O
we call it the derivative off at x 0 , and denote it f'(xo).
Thus, f'(x ) lim f (xo + h) - f (xo) if this limit exists.
O = h-+O hTheorem 6.1.7 (Sequential Criterion for Differentiability)
Suppose f : D(f) --t IR and xo is an interior point of D(f). Then f is
differentiable at xo with derivative f'(xo) iff '</sequences {xn} in D(f)-{xo}
sue h th a t Xn --t xo, f (xn) - f (xo) --t f'( xo ).
Xn -Xo
Proof. This is a trivial application of Theorem 4.1.9 to Definition 6.1.6. •Since the definition of derivative involves the concept of limit, the concept
of continuity cannot be far away. It is natural to ask whether there is a relation
between these two concepts: continuity of a function f at x 0 and differentiability
off at xa. The following theorem establishes this relationship.
Theorem 6.1.8 (Differentiability Implies Continuity) If f is differen-
tiable at xo, then f is continuous at xo.
Proof. Suppose f is differentiable at xo. Then, for all x E D(f) - { x 0 },f(x) = f(x) - f(xo) (x - xo) + f(xo).
x - xoThus, by the algebra of limits,1. im f( x ) = 1. im f(x) - f(xo) 1. Im ( x - Xo ) + 1. Im f( Xo )
x-+xo x-+xo x - xo x-+xo x-+xo= f'(xo)(xo - xo) + f(xo) = 0 + f(xo) = f(xo).
Thus, xlim -+xo f(x) = f(xo), which means that f is continuous at x 0. •