  H\"older functions represent mathematical models of nonlinear physical
phenomena. This work investigates the general conditions of existence of
fractional velocity as a localized generalization of ordinary derivative with
regard to the exponent order. Fractional velocity is defined as the limit of
the difference quotient of the function's increment and the difference of its
argument raised to a fractional power. A relationship to the point-wise
H\"older exponent of a function, its point-wise oscillation and the existence
of fractional velocity is established. It is demonstrated that wherever the
fractional velocity of non-integral order is continuous then it vanishes. The
work further demonstrates the use of fractional velocity as a tool for
characterization of the discontinuity set of the derivatives of functions thus
providing a natural characterization of strongly non-linear local behavior.
Finally the equivalence with the Kolwankar-Gangal local fractional derivative
is investigated.
