  We find an equivalent condition for a continuous vector-valued path to be
Lebesgue equivalent to a twice differentiable function. For that purpose, we
introduce the notion of a $VBG_{{1/2}}$ function, which plays an analogous role
for the second order differentiability as the classical notion of a $VBG_*$
function for the first order differentiability. In fact, for a function
$f:[a,b]\to X$, being Lebesgue equivalent to a twice differentiable function is
the same as being Lebesgue equivalent to a differentiable function with a
pointwise Lipschitz derivative. We also consider the case when the first
derivative can be taken non-zero almost everywhere.
