1) The function can have a discontinuity (blue line); 2) The function can have a cusp (red line).
Work Step by Step
1) The function can be discontinuous for some $a$. Since all differentiable functions are continuous, if the function has a discontinuity, then it is not differentiable (blue line). 2) The function can have a cusp at some $a$. In that case at $x=a$ the function has two different tangents with different slopes and because the derivative is unique, at the cusp the function is not differentiable (red line).