Sup

I see it in the Legendre transformation, which states:

f*(p) = sup(p*x-f(x))

What is the "sup" function? What does it do? How is it used? What is the point of using it? How does the Legendre transformation work? Can you please provide examples (preferably with functions that variables can be plugged into)?

What does p represent? I know that in transformations, f(x) is usually mapped onto a different variable, but in this particular transformation, doesn't really seem to be doing anything to f(x), because it's external to f(x).

Also, if the sup function is the supremum function, does that imply that the curve is full of maxima? This is really confusing to me, and I feel like I need to understand this before I can start to grasp thermodynamics.

 Unrelated question

Out of curiosity: If I differentiate x(t) = cos(ω*t), ω>1, an infinite number of times, does it eventually diverge? In other words, if x'(t) is the velocity, x''(t) is the acceleration, x'''(t) is the jerk, ... and the infinite derivative is the infinite change in some quantity of motion, does that imply that a simple oscillator can have an infinitely large change in some quantity describing the motion?