What are non differentiable points for a graph?

1 Answer
Jul 24, 2014

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.