Let f 1 1 rightarrow R be continuous and differentiable on 1

Let f: [-1, 1] rightarrow R be continuous, and differentiable on (-1, 1)\\{0}. Assume that lim_x rightarrow 0 f\'(x) = l. Prove that f is differentiable at 0 and that f\'(0) = l.

Solution

Definition of derivative : f \'(x) =Lim h->0 [ f (x+h) -f(x) ] / h --------(1)

given: Lim x-> 0 f \'(x) =l ----(2)

LHS of (2) is Lim x-> 0 Lim h->0 [ f (x+h) -f(x) ] / h (using (1))

interchangng the limits

Lim h->0 Lim x->0 [ f (x+h) -f(x) ] / h

= Lim h->0 [ f (0+h) -f(0) ] / h

= Lim h->0 [ f (h) -f(0) ] / h

LHS = f \'(0)= l = RHS of (2)

ie f \'(0)= l