# Wronskians and linear dependence

If $f_1,\dots,f_k:I\subseteq\Bbb{R}\to\Bbb{R}$ are sufficiently differentiable, their Wronskian is defined as

$W(f_1,\dots,f_k)=\mathrm{det}\begin{bmatrix} f_1&f_2&\dots&f_k\\f_1'&f_2'&\dots&f_k'\\ \vdots&\vdots&\ddots&\vdots\\ f_1^{(k-1)}&f_2^{(k-1)}&\dots&f_k^{(k-1)}\end{bmatrix}$

Obviously, if $f_1,\dots,f_k$ are linearly dependent, then their Wronskian vanishes identically on $I$, and the converse is true if the functions are analytic. Find non-analytic functions for which the converse fails.

1. The functions $x^2$ and $x^2\,\mathrm{sgn}(x)$ do the trick as well.