I need an example of a function that is continuous at some x0 but not differentiable at x0.
is f(x) = |x| a right answer?
a uniformly continuous function which is nowhere differentiable. In fact, given a continuous function you may construct a continuous function such that for some fixed but arbitrary and which is nowhere differentiable. Your answer also shows that given a function which is differentiable at that the function is not differentiable at giving and infinitely weaker form of the aforementioned theorem.