For the eigenvalue function on symmetric matrices, we have gathered a number of it’s properties.We show that this map has the properties of continuity, strict continuity, directional differentiability, Frechet differentiability, continuous  differentiability. Eigenvalue function will be extended to a larger set of matrices and then the listed properties will prove again.

Let (E, ‖·‖E) be a symmetric space and let Y ⊂ X be a nonempty subset. For x ∈ X denote PY (x) = {y ∈ Y : ‖x− y‖ = dist(x, Y )}. Any element y ∈ PY (x) is called a best approximant in Y to x. A nonempty set Y ⊂ X is called proximinal or set of existence if PY (x) 6= ∅ for any x ∈ X. A nonempty set Y is said to be a Chebyshev set if it is proximinal and PY (x) is a singleton for any x ∈ E. A sym...

Approximately continuous functions were first introduced in connection with problems of differentiation and integration. The main idea of the approximate continuity of a function f is that the continuity conditions are true not necessarily everywhere but only almost everywhere with respect to some measure, e.g., Borel measure or Lebesgue measure. At the same time, it is known that functions tha...

Varieties of the Fan Theorem have recently been developed in reverse constructive mathematics, corresponding to different continuity principles. They form a natural implicational hierarchy. Some of the implications have been shown to be strict, others strict in a weak context, and yet others not at all, using disparate techniques. Here we present a family of related Kripke models which suffices...