Derivatives -up to the third order

[Camillo]

Calculate the derivatives, up to the third order , of the function $u(x) = x |x| , x in RR $.

[Camillo]

I write in a more compact way :

$f'(x) = 2|x| $
$f''(x) =2 sign(x) $
$f'''(x) $ is not simply $0$(this would just be the "classical derivative") , but to take into account the discontinuity in $x=0 $ with a hop of $4 $ we conclude that $ f'''(x) = 4 delta(x) $ (and this is the derivative in the sense of distributions) , being $delta(x) $ the Dirac function .

[gugo82]

"Camillo":Calculate the derivatives, up to the third order , of the function $u(x) = x |x| , x in RR $.

Since $f$ is an odd function of the variable $x$, we can calculate the derivatives up the third order for positive values of $x$ and then obtain their expression for negative values of the variable by a suitable continuation (even or odd depending upon the order of the derivative); at least we can discuss about the chance to continuate these functions on $0$ saving their continuity.

We have for $x>0$, $f(x)=x^2 quad => quad f'(x)=2x quad => quad f''(x)=2 quad => quad f'''(x)=0$.
Since $f$ is an odd function, for $x<0$ we can continuate the derivatives on the whole negative real semiaxis by put $AA i in {1,2,3}, AA x<0, f^((i))(x)=(-1)^(i+1)*f^((i))(-x)$: then we got:

$f'(x)=\{(2x, " if " x>0), (-2x, " if " x<0):} quad$;
$f''(x)=\{(2, " if " x>0), (-2, " if " x<0):} quad$;
$f'''(x)=0 quad$.

The first and the third derivatives of $f$ can be extended over $0$ by defining $f'(0)=0=f''(0)$ with continuity, but we can't do the same with $f''$ since $lim_(x to 0^-)f''(x)!=lim_(xto 0^+)f''(x)$. This implies that $f in C^1(RR)$ but it's not of a higher class of differentiability over the whole real axis (infact $f$ is only locally of class $C^oo$).