1549901369-Elements_of_Real_Analysis__Denlinger_

302 Chapter 6 • Differentiable Functions

Therefore, f is differentiable at 0, and f'(O) = 0. D

Another caution: Although differentiability off implies continuity off,
it does not imply continuity off'. In fact, the function f defined in Example
6.1.10 is differentiable at 0, but f' is not continuous there. (See Exercise 6.2.17.)

ONE-SIDED DERIVATIVES

Definition 6.1.11 Suppose f : 1J(f) --. ~-

(a) Suppose 1J(f) includes an interval of the form (xo-8, xo], for some 8 > 0.

Then f is differentiable from the left at x 0 if the limit lim f(x) - f(xo)
x-+x() X - Xo
exists (i.e., is finite). If this limit exists, we call it the derivative from the
left of f at xo, and denote it f'_ ( xo).

Thus, f'_(xo) = lim f(x) - f(xo) if this limit exists.

x -+x() X - Xo

(b) Suppose 1J(f) includes an interval of the form [x 0 , x 0 + 8), for some 8 >

0. Thenf is differentiable from the right at x 0 ifthe limit lim f(x) - f(xo)
   x-+x;i X - Xo
   exists (i.e., is finite). If this limit exists, we call it the derivative from the
   right off at xo, and denote it f~(xo).

Thus, f~(xo) = lim f(x) - f(xo) if this limit exists.

x-+x;i X - Xo

Example 6.1.12 In Example 6.1.4 we showed that for the function f(x) = lxl,
f'_ (O) = -1, while f~ (0) = l.

Theorem 6.1.13 Suppose f : 1J(f) --. ~ and xo is an interior point of 1J(f).
Then f' ( xo) exists {::} both f '_ ( xo) and f ~ ( x 0 ) exist and are equal.

Proof. Exercise 11. •

*Theorem 6.1.14 (a) If 38 > 0 3 f is differentiable on (x 0 - 8 ,x 0 ) and
continuous from the left at Xo, and lim f' ( x) exists, then f' ( x 0 ) exists and
x-+xQ
equals lim f' (x).

•An asterisk with a theorem, proof, or other materia l in this chapter indicates that the item
is challenging and can be omitted, especially in a one-semester course.