Smoothness
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?

Re: Smoothness
(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, singlevalued, 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.)

Re: Smoothness
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. 
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