What is the definition of a cusp? So there are 3 situations that a function is not differentiable 1. a vertical tangent, 2. a discontinuity and 3. at a cusp. Consider the function $f(x)=e^{-|x|}$ at the point $x=0$. It is continuous here but its first derivative is not does, and therefore it cannot be differential here can it? But I have read that a cusp is when the limit of the first derivative must tend to $+\infty$ when approaching the point from one direction and $-\infty$ when from the other. In this case it tends to $+1$ and $-1$ which should mean that it does not have a cusp here and does not fall into one of the non differentiable categories. But it is not differentiable, so is this definition of a cusp wrong, if so what is the actual definition? thanks.  
 A: Maxima or minima for functions occur at three types of points:
a. Stationary points where the derivative equals zero.
b. Corners where the left sided derivative does not equal the right sided derivative - for example, $y = |x|$.  This is a perfect example of a corner.
c. Cusps, which are infinitely sharp corners.  A cusp is a point where you have a vertical tangent, but with the following property: on one side the derivative is $+\infty$, on the other side the derivative is $-\infty$. The paradigm example was stated above: $y = x^{2\over3}$. The limit of the derivative as you approach zero from the left goes to $-\infty$. The limit of the derivative as you approach zero from the right goes to $+\infty$. At $x = 0$, there is a vertical tangent.  
Vertical tangents exist at cusps (when describing functions). But a vertical tangent itself is not a cusp. The curve $y = x^{1\over3}$ has a vertical tangent at the origin but the there is no cusp: the curve is smooth, there is no pointy needle-like behavior to it.
A: A cusp is understood to be "pointy", such as the graph of the function $y=\sqrt[3]{x^2}$ at $x=0$.
See also cusp catastrophe
Or in other contexts, consider the upper half plane $\mathbb H$ modulo the action of $PSL(2,\mathbb Z)$: It is essentially a curved polygon with sided glued together, thereby producing two non-pointy corners and one cusp (because edges are glued together that have a common tangent at the vertex, just like the function graph of the first example
