  We define a deterministic integral with respect to irregular paths as a limit
of standard line integrals and completely describe a class of all paths for
which this integral exists for functions with H\"older exponent in the range of
(0,1]. With the developed integral calculus, we study the existence, uniqueness
and continuity of solution of time- and path-dependent ordinary differential
equations driven by irregular paths in traditional H\"older spaces. These
results can be viewed as a supplement to the Young-Lyons-Gubinelli integration
theory, in particular, for H\"older exponents less than 1/3.
