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 ] → X , being Lebesgue equivalent to a twice differentiable function is the same as being Lebesgue equivalent to a differentiable function g with a pointwise Lipschitz derivative such that g ″ ( x ) exists whenever g ′ ( x ) ≠ 0 . We also consider the case when the first derivative can be taken non-zero almost everywhere.