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 X and y_0 (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 functions defined on separable Hilbert spaces and Asplund spaces.