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 $f{g}^{\prime }-f^{\prime }g=0$ 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.