Definition of cusp or corner

Question

I've seen many times the terms like "cusp" and "corner" in the calculus books but I want a formal and rigorous definition for it . I know $f(x) = |x^2 - 1|$ has a cusp at $x = 1$ but I'm in doubt about $g(x) = \sqrt[3]{x^2}$ at $x= 0$ .

Answer 1

At an elementary stage, when one talks about a cusp or a corner of a curve $h(t)$ at $x_0$, it would mean that:

$\lim_{t\rightarrow 0}\frac{h(x_0+t)-h(x_o)}{t} $ doesn't exist and in particular it is different when $t$ approaches $0$ from the left and from the right.

It is customary to talk of cusps to curves that are otherwise smooth (infinitely differentiable) and while continuous at $x_0$ they have a singularity.

As Arnaud Mortier noted in the comments, in more sophisticated settings (e.g. algebraic or differential geometry) cusp has a more complex and precise definition.

Answer 2

The example that you give, of the curve $y^3=x^2$, corresponds to what is called a regular cusp or ordinary cusp, in the sense that it is the least degenerate from the point of view of singularity theory. This leads us to:

Definition 1 (used by some authors): a cusp is precisely a point where a curve is equivalent to $y^3=x^2$ via a smooth local change of variables.

Definition 2 (more general): A piecewise smooth curve in, say, a euclidean space: $$f: \mathbb{R}\to\mathbb{R}^n$$ has a cusp at a point $p=f(x)$ if there is $\varepsilon>0$ such that $f$ is differentiable on both intervals $(x-\varepsilon, x)$ and $(x, x+\varepsilon)$, and if $$\lim_{h\to 0^+} f_1'(x-h)=\lim_{h\to 0^+} f_1'(x+h)$$ where $f_1'$ denotes the unit tangent vector to the curve.

In words, the curve leaves the point in the same direction as it has reached it.