  Holderian functions have strong non-linearities, which result in
singularities in the derivatives. This manuscript presents several
fractional-order Taylor expansions of H\"olderian functions around points of
non- differentiability. These expansions are derived using the concept of a
fractional velocity, which can be used to describe the singular behavior of
derivatives and non-differentiable functions. Fractional velocity is defined as
the limit of the difference quotient of the increment of a function and the
difference of its argument raised to a fractional power. Fractional velocity
can be used to regularize ordinary derivatives. To this end, it is possible to
define regularized Taylor series and compound differential rules. In particular
a compound differential rule for Holder 1/2 functions is demonstrated. The
expansion is presented using the auxiliary concept of fractional co-variation
of functions.
