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 Föllmer's pathwise Itō calculus. Our construction of the latter class of functions relies on an extension of the Doss--Sussman method to a class of nonlinear Itō differential equations for the Föllmer 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.