Function $f(x)$ is differentiable at a point P, when there exist a unique tangent at the point P. The common situations in which a function would not be differentiable at a point – vertical tangent lines, discontinuities and sharp turns in the graph.
Consider the function $f(x)$ = $|x|$. The graph of this function is given below
The graph shows that $f(x)$ is not differentiable at $x = 0$ as $f(x)$ has sharp edge at $x=0$. The function is continuous at $x =0$ but it is not differentiable there. This clearly says that continuity is not a strong enough condition to guarantee differentiability. On the other hand, when the function is differentiable at a point then it should be continuous at that points. i.e, if a function $f$ is differentiable at $x=c$, hence $x$ is continuous at $x=c$.