Construction of Intermediate Differentiable Functions

Abstract

For given upper and lower semicontinuous real-valued functions g and h, respectively, defined on a closed parallelepiped X in ℝn and such that g(x) < h(x) on X and points x0 ϵ X and y0 ϵ (g(x0),h(x0)), we construct an infinitely differentiable function f : X → ℝ such that f(x0) = y0 and g(x) < f(x) < h(x) on X. We also present similar structures for functions defined on separable Hilbert spaces and Asplund spaces.