[For the entire collection see Zbl 0556.00005.]
Let \(R_ k\) be the k-dimensional Euclidean space, \({\mathcal X}\) be the space of all nondegenerate closed intervals in \(R_ k\) with the metric \(\rho (I,J)=m(I\Delta J),\) where \(I,J\in {\mathcal X},\) m is the Lebesgue measure in \(R_ k\) and \(I\Delta J\) is the symmetric difference of I and J. For \(I\in {\mathcal X}\) d(I) will denote the diameter of I. An interval function \(F: {\mathcal X}\to R\) is called continuous iff it is continuous on \(({\mathcal X},\rho).\) Let \({\mathfrak e}=(e_ 1,...,e_ k),\) where \(e_ i=1\) for each \(i=1,...,k,\) \(R^+_ k=\{h_ 1,...,h_ k):\) for each \(i\in \{1,...,k\}:\) \(h_ i>0\}\) and \({\mathfrak i}=(i_ 1,...,i_ k)\) be any point for which \(i_ j\in \{-1,1\}\) for each \(j\in \{1,...,k\}.\) Let \(F:\quad {\mathcal X}\to R\) and \(X\in R_ k\). Then the upper \({\mathfrak e}\)- strong derivative \(\bar F^+_ s(X)\) of F at X, and the upper strong derivative \(\bar F_ s'(X)\) of F at X are defined as follows: \[ \bar F^+_ s(X) = \limsup_{d(I) \to 0} \left\{\frac{F(I)}{m(I)}: I=<X,Y>, \text{ \(X\) and \(Y\) are main opposite corners of\(I\) and }Y\in X+eR^+_ k \right\} \] and \[ \bar F_ s'(X)= \limsup_{d(I) \to 0} \left\{\frac{F(I)}{m(I)}: X\in I\right\}, \] respectively. A function \(f: R_ k\to R\) will be called lower \({\mathfrak i}\)-semicontinuous at X iff for each \(a\in R\) the inequality \(a<f(X)\) implies the existence of a \(Y\in R^+_ k\) such that \(f(<X,X+{\mathfrak i}Y>)\subset \{Z\in R_ k: f(Z)>>a\}.\)
There is proved the following theorem: a) For any continuous function \(F: {\mathcal X}\to R\) the function \(\bar F^+_ s\) is a limit of a nonincreasing sequence of lower semi-continuous functions. b) For any subadditive function \(F: {\mathcal X}\to R\) \(\bar F_ s'\) is a limit of a nonincreasing sequence of functions which are lower \({\mathfrak i}\)- semicontinuous at any X, where \({\mathfrak i}\) depends on X and on the function.