Question

prove : if f(x) is differentiable at x, then f(x) is continuous at x
or vice versa

Answer 1

Differentiability implies continuity
But continuity doesn't necessarily imply differentiability

Answer 2

...still, how to prove it?

Answer 3

If a function f(x) is differentiable on x = a
then the term
lim x->a (f(x) - f(a)) / (x-a) is well defined.

then

lim x->a (f(x) - f(a)) = lim x->a (x-a) (f(x) - f(a))/(x-a)
=lim x->a (x-a) lim x->a (f(x) - f(a))/(x-a) = 0 * f'(a) = 0

lim x->a (f(x) - f(a)) = 0
=> lim x->a (f(x) = f(a)) (which implies continuity)

But reverse may not apply, for example
f(x) = |x| is not differentiable at x=0 though continuous

Answer 4

the other example would be
x sin(1/x) at x = 0
http://en.wikipedia.org/wiki/Weierstrass_function