Definition : A function f ( x ) is called differentiable at x = a if f ′ ( x ) exists & f ( x ) is called differentiable onto an interval if the derivative present for each of the point in that interval.

The next theorem illustrates us a very nice relationship among functions which are continuous & those that are differentiable.

Theorem: If  f ( x ) is differentiable at x = a then f ( x ) is continuous at the point x = a.

Note  as well that this theorem does not work in reverse order .  Assume f ( x ) = |x| and take a look at,

So,f ( x ) = xis continuous at x =0 to sketch the graph of a function.