
“real: chapter_06” — 2011/5/22 — 23:14 — page3—#3
Differentiation 6-3
6.2 DEFINITION OF DERIVATIVE, EXAMPLES AND
ARITHMETIC RULES
Definition 6.2.1 Let S ⊂ R. Let L(S) denote the set of limit points of
S and x ∈ S ∩ L(S). We say that f is differentiable at x if
f
(x) = lim
t→x
f (t) − f (x)
t −x
exists as a real number. We also call f
(x) as the derivative of f at x. If
f is differentiable at all points of S, then we say that f is differentiable
on S.
Note 6.2.2 If S is an interval and x is not an end point of S, then t can
approach x with either of the conditions t > x or t < x. This f
(x) will
be called the two-sided derivative of f at x . For similar reasons, ...