As
in case of continuity, at the end points a and b, we take the right hand limit and left hand
limit, which are nothing but left hand derivative and right hand derivative of the function
at a and b respectively Similarly, a function is said to be differentiable in an interval
(a, b) if it is differentiable at every point of (a, b) Rationalised 2023-24
 MATHEMATICS
120
Theorem 3 If a function f is differentiable at a point c, then it is also continuous at that
point Proof Since f is differentiable at c, we have
( )
( )
lim
( )
x
c
f x
f c
f c
x
c
→
−
=
′
−
But for x ≠ c, we have
f (x) – f (c) =
( )
( )