  We construct rich vector spaces of continuous functions with prescribed
curved or linear pathwise quadratic variations. We also construct a class of
functions whose quadratic variation may depend in a local and nonlinear way on
the function value. These functions can then be used as integrators in
F\"ollmer's pathwise It\=o calculus. Our construction of the latter class of
functions relies on an extension of the Doss--Sussman method to a class of
nonlinear It\=o differential equations for the F\"ollmer integral. As an
application, we provide a deterministic variant of the support theorem for
diffusions. We also establish that many of the constructed functions are
nowhere differentiable.
