Question

What are non differentiable points for a graph?

Answer 1

Since a function that is differentiable at `a` is also continuous at `a`, one type of points of non-differentiability is discontinuities .

On the other hand, if the function is continuous but not differentiable at `a`, that means that we cannot define the slope of the tangent line at this point. This can happen in essentially two ways:
1) the tangent line is vertical (and that does not have a slope)
2) the difference quotient `(f(x)-f(a))/(x-a)` whose limit at `a` defines the derivative has two different one-sided limits at `a`, resulting in two half-tangents. We call this situation a "cusp".
See this video on differentiability for details and pictures.