# What are non differentiable points for a graph?

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 $\frac{f \left(x\right) - f \left(a\right)}{x - a}$ 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.