Continuity of set-valued maps is hereby revisited: after recalling some basic concepts of variational analysis and a short description of the State-of-the-Art, we obtain as by-product two Sard type results concerning local minima of scalar and vector valued functions. Our main result though, is inscribed in the framework of tame geometry, stating that a closed-valued semialgebraic set-valued ma...

The paper gives a brief account of the spaces of interval functions defined through the concepts of H-continuity, D-continuity and S-continuity. All three continuity concepts generalize the usual concept of continuity for real (point valued) functions. The properties of the functions in these new spaces are discussed and investigated, preserving essential properties of the usual continuous real...

A Qk(R ) -valued function is essentially a rule assigning k unordered and non necessarily distinct elements of R to each element of its domain set A ⊆ R . For a Qk(R) valued function f we construct a decomposition into k branches that naturally inherit the regularity properties of f . Next we prove that a measurable Qk(R ) -valued function admits a decomposition into k measurable branches. An e...

It is useful to treat real-valued functions (or complex-valued functions, or vector space-valued functions) as elements of a vector space, so that the tools from linear algebra can be applied. Given a set X one may consider the vector space R of all real-valued functions with domain X. If X is finite, say with n elements, then this is just the familiar vector space R. The more interesting examp...

CMAC (Albus, 1975b) is well known as a good function approximator with local generalization abilities. Depending on the smoothness of the function to be approximated, the resolution as the smallest distinguishable part of the input domain, plays a crucial role. If the usually used binary quantizing functions are dropped in favor of more general, continuous-valued functions, this drawback can be...

Let G be a topological group with the identity element e. Given a space X, we denote by Cp(X,G) the group of all continuous functions from X to G endowed with the topology of pointwise convergence, and we say that X is: (a) G-regular if, for each closed set F ⊆ X and every point x ∈ X \ F , there exist f ∈ Cp(X,G) and g ∈ G \ {e} such that f(x) = g and f(F ) ⊆ {e}; (b) G-regular provided that t...