Hi, Function y = |x| belongs to the class C^0, but does not belong to C^1. And what function belongs to C^1, but does not belong to C^2?

(1). given two curves, jointed at at point, then the smoothness is C^0. Zeroth order continuous, I guess. (2). if you can also find the first order derivative at that point, single-valued, then you have C^1. That is, both the function and its derivative are continuous. (3). So, function x will be continuous, and also has continuous first derivative, and is belong to C^1. (1st derivative is Continuous.)

f(x) = abs(x) is C^0 over [-2,+2]

because

f'(x) = -1 over [-2,0[

f'(x) = +1 over ]0,+2]

and it's undefined at 0.

F(x) = int from{-2} to{x} of (abs(t)) dt = x*abs(x)/2 + 2

is C^1 since its first derivative is abs(x) wich is C^0.