Second order differentiability of paths via a generalized 12 -variation

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 VBG1/2 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. © 2007 Elsevier Inc. All rights reserved.