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.

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Wronskians and linear dependence

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