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?
Yes. Or a uniformly continuous function which is nowhere differentiable. In fact, given a continuous function $\displaystyle f:I\to\mathbb{R}$ you may construct a continuous function $\displaystyle g:I\to\mathbb{R}$ such that $\displaystyle \|f-g\|_{\infty}<\varepsilon$ for some fixed but arbitrary $\displaystyle \varepsilon>0$ and which is nowhere differentiable. Your answer also shows that given a function $\displaystyle f$ which is differentiable at $\displaystyle x_0$ that the function $\displaystyle f(x)+|x-x_0|$ is not differentiable at $\displaystyle x=x_0$ giving and infinitely weaker form of the aforementioned theorem.