Classification of non-differentiability points

When $f$ is continuous at $x_0$ but not differentiable, the behaviour of the two one-sided derivatives distinguishes three cases. This table is the guiding scheme for classifying any non-differentiability point.

In brief — Classification of non-differentiability points

Let $f$ be continuous at $x_0$ but not differentiable at $x_0$. Depending on the limits $f'_\pm(x_0)$:

Case	$f'_-(x_0)$ and $f'_+(x_0)$	Graphical behaviour
Corner point	finite, distinct	two half-tangents, sharp angle
Cusp	infinite, opposite signs	two “opposite” vertical half-tangents
Inflection with vertical tangent	infinite, same sign	single, vertical tangent

Links

Topics: Derivatives
Concepts: Cusp · Corner point · Non-differentiability point · Vertical tangent
Methods: Non-differentiability point classification
Skills: Reasoning by cases