6.1 The Derivative and Differentiability 301Theorem 6.1.8 says that differentiability is "stronger" than conti-
nuity; that is, differentiability implies continuity. Differentiability of f at xo
implies continuity off at xo.
Caution: Continuity does not imply differentiability. There are
functions f that are continuous at x 0 but not differentiable at xo. The ab-
solute value function f(x) = lx l at 0 is one such example. Its continuity was
established in Example 5.1.9 (a), and its nondifferentiability in Example 6.1.4.
The following example provides another illustration of this happening.
Example 6.1.9 The function f(x) = { xsin ~ ~f x IO,} is continuous at 0,
0 if x = 0
but not differentiable there. (See graph in Example 4.2.21.)
Proof. (a) Continuity at 0:x-+O lim f ( x) = x-+O lim (x sin 1.) x
= 0 (See Example 4.2.21.)
= f(O).Since lim f(x) = f(O), f is continuous at 0.
x-+O(b) Nondifferentiability at 0:. f ( x) - f ( 0). x sin 1. - 0
bm bm x
x-+O X - 0 x-+O X - 0
x sin 1.
lim __ x
x-+O X
= lim sin 1., which does not exist (Example 4.1.12).
x-+O x
Therefore, f is not differentiable at 0. 0
The following example is an interesting variation of Example 6.1.9.1 Th f. f ( ) { x
2
Examp e 6.1.10 e unct10n x = sin 1. x if x I^0 } is. d·cr iueren t. ia bl e
0 if x=O
at 0, and f' (0) = 0.
Proof.
1. f(x) - f(O)
im ----'--'----
x-+ O X - 0
x^2 sin1. - 0
lim x
x-+O X - 0
x^2 sin 1.
= lim x
x-+O X
= lim xsin 1. = 0. (Example 4.2.21).
x-+O x