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 $R^n$ and such that $g(x) < h(x)$ on $X$ and points $x_0 \in X$ and $y_0 \in (g(x_0), h(x_0))$, we construct a smooth function $f : X \rightarrow R$ such that $f(x_0) = y_0$ and $g(x) < f(x) < h(x)$ on $X$. We also present similar constructions for functions defined on separable Hilbert spaces and Asplund spaces.