= Vanishing Wronskian does not imply linear dependence
Two <differentiable functions> can be <linearly independent> on a connected <interval> while their <Wronskian> vanishes everywhere. For example, let $f(x)=(\max\{-x,0\})^2$ and $g(x)=(\max\{x,0\})^2$. Both have a continuous first <derivative>, and their values and first derivatives vanish at zero. On either side of zero one of the functions is identically zero, so $fg'-f'g=0$ everywhere. Evaluating a constant <linear combination> $Af+Bg=0$ at $-1$ and $1$ forces $A=B=0$.
The obstruction is that the two halves supply independent information without simultaneous nonzero value-and-derivative data at any point. For solutions of a common regular <second-order linear differential equation>, the <uniqueness theorem for ordinary differential equations> removes this possibility; <Abel's identity> gives the zero-or-nowhere-zero alternative for their <Wronskian>.
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