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Uniformly Antisymmetric Functions And K5, Krzysztof Ciesielski
Uniformly Antisymmetric Functions And K5, Krzysztof Ciesielski
Faculty & Staff Scholarship
A function f from reals to reals (f:R-->R) is a uniformly antisymmetric function if there exists a gage function g:R-->(0,1) such that |f(x-h)-f(x+h)| is greater then or equal to g(x) for every x from R and 0R-->N, (see [K. Ciesielski, L. Larson, Uniformly antisymmetric functions, Real Anal. Exchange 19 (1993-94), 226-235]) while it is unknown whether such function can have a finite or bounded range. It is not difficult to show that there exists a uniformly antisymmetric function with an n-element range if and only if there exists a …
Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous, Krzysztof Ciesielski
Cardinal Invariants Concerning Functions Whose Sum Is Almost Continuous, Krzysztof Ciesielski
Faculty & Staff Scholarship
Let A stand for the class of all almost continuous functions from R to R and let A(A) be the smallest cardinality of a family F ⊆ R R for which there is no g: R → R with the property that f + g ∈ A for all f ∈ F. We define cardinal number A(D) for the class D of all real functions with the Darboux property similarly. It is known, that c < A(A) ≤ 2 c [10]. We will generalize this result by showing that the cofinality of A(A) is greater that c. Moreover, we will show that it is pretty much all that can be said about A(A) in ZFC, by showing that A(A) can be equal to any regular cardinal between c + and 2c and that it can be equal to 2c independently of the cofinality of 2c . This solves a problem of T. Natkaniec [10, Problem 6.1, p. 495]. We will also show that A(D) = A(A) and give a combinatorial characterization of this number. This solves another problem of Natkaniec. (Private communication.)