Wronskian of x and |x|

Question

I am asked to find the Wronskian of $x$ and $|x|$ in $[-1,1]$.
But $|x|$ is not differentiable at $x=0$.
How do I calculate the Wronskian of such non-differentiable functions?
Could I do this: $W(x,|x|)|_{x\in[-1,0)}+W(x,|x|)|_{x\in(0,-1]}$ because $x$ and $|x|$ both are $0$ at $x=0$.

Answer 1

Wronskian is defined for differentiable functions. To define $W(f_1, ..., f_n)(x)$ on some interval $I$, functions $f_1, ...f_n$ must be at least $n-1$ times differentiable on that intreval.

Since $|x|$ is not differentiable on $[-1,1]$, so the Wronskian is not defined for $x$ and $|x|$ on that interval.

More precisely the Wronskian is not defined at $x=0$ and for $x \neq 0$ the Wronskian is

$$ W(x, |x|)(x) = \begin{vmatrix} x & |x| \\ 1 & \frac{|x|}{x}\\ \end{vmatrix} = |x| - |x| = 0. $$