Consider the function $y=|x|$ in the interval $[-1, 1]$. In this interval, the function is
$y = |x| = \begin{Bmatrix} x & x\geqslant 0 \\ -x& x< 0 \end{Bmatrix}$
$y' = \begin{Bmatrix} 1 & x\geqslant 0 \\ -1& x< 0 \end{Bmatrix}$
continuous but not differentiable at x = 0
http://math.stackexchange.com/questions/991475/why-is-the-absolute-value-function-not-differentiable-at-x-0