My professor said in class:

Can someone please give me an example of a case where the limit does not need to exist at x=a for the function to have a derivative at x=a ?The derivative of f at x=a is f'(a) = [f(a+h) - f(a) ] /h (and said that this Limit does not need to exist for the derivative to exist). (Then he said) If f'(a) exists, then f is differentiable at x=a

If f'(a) exists, does that mean f is continuous on x=a? ( I think yes) But this is not the other way around?

Thank you in advance.