Construction of intermediate differentiable functions - Maslyuchenko, V. K.; Mel'nik, V. S. (Ukrains’kyi Matematychnyi Zhurnal, 2018)
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$f : X → R$ such that $f(x_0) = y_0$ and $g(x) < f(x) < h(x)$ on $X$. We also present similar constructions for functionsdefined on separable Hilbert spaces and Asplund spaces.