cusp vs. corner? or both? I searched through books and internet and they all have general definitions of them as follows:
Cusp: where the slope of the tangent line changed from -infinity to +infinity (or the other way around)
Corner: left-sided and right-sided derivatives are different.
And I saw a problem which was asking if there is a corner or a cusp given a graph. The graph looked like:
f(x)=-x, if x<0
=sqrt(x), if x>=0

So in short, one branch was straight, and another branch was curved. I know the point where x=0 is not differentiable. But would it be considered a corner or a cusp? In my opinion, it should be a corner because it does not change from -infinity to +infinity. However, while I was searching, I saw an example of graph that looks similar to that, and the website was calling it a cusp (sorry I cannot find the image anymore).
Also, this is another question, but if a cusp have a slope of either -infinity or +infinity, wouldn't it be a subcategory of vertical tangent?
 A: A corner point has two distinct tangents. A cusp has a single one which is vertical.
A: The first "curve" has DE $ dy/dx= \cos y/ \cos x,\quad  y = \tan^{-1} (\tan x ) $  with slope $+1$ everywhere. A regular continuous curve.
In second curve with a corner it has first degree contact i.e., same $(x,y)$, first and second degree values (slope,curvature) can be different. Here wave equation  $ y= \cos^{-1} (\cos x), $ there is change in slope at the corner point $x=\pi$. Slope goes from $1$ to $-1$ via $0$ with tangent parallel to x-axis.

Second example below of curve with a corner DE $ y^{''2} +1- y^2 =0 $ is a periodic curve.  BC $ (0,-\frac12) $. At the corner $(x \approx 1.81,y=1), y^{'}= \pm 1.59, y^{''}=0 $.

A cusp has first degree contact, same slope of infinite curvature whose sign changes at cusp location and the slope also passes through either zero or infinity value. Examples are point of contact of a circle when a cycloid type curve is produced by rolling circles on a straight line or another circle. Typically, last point of tracks when direction changes in an automobile in reverse gear.
There are curves that are synthesized from individual curves like discontinuous slope electrical wave-forms (square, trapezium, saw-tooth etc). Its Fourier components are evaluated and next, at discontinuity points slope and $y$ values are averaged out.
As to your last question. A circle of a circular toroid has finite curvature at point of vertical tangency. A cusp has infinite curvature at vertical tangent, but not a sub-category.
