On the family of functions with a closure of its graph of measure zero
On the maximum of generalized Darboux functions
Pointwise regularity associated with function spaces and multifractal analysis
The purpose of multifractal analysis of functions is to determine the Hausdorff dimensions of the sets of points where a function (or a distribution) f has a given pointwise regularity exponent H. This notion has many variants depending on the global hypotheses made on f; if f locally belongs to a Banach space E, then a family of pointwise regularity spaces are constructed, leading to a notion of pointwise regularity with respect to E; the case corresponds to the usual Hölder regularity, and...
Prevalence of some known typical properties.
Products of quasi-continuous functions
Relativization of some aspects of the theory of functions of bounded variation [Book]
Smooth Cantor functions
We characterise the set on which an infinitely differentiable function can be locally polynomial.
Solutions with big graph of iterative functional equations of the second order.
Solutions with big graph of iterative functional equations of the first order
We obtain a result on the existence of a solution with big graph of functional equations of the form g(x,𝜑(x),𝜑(f(x)))=0 and we show that it is applicable to some important equations, both linear and nonlinear, including those of Abel, Böttcher and Schröder. The graph of such a solution 𝜑 has some strange properties: it is dense and connected, has full outer measure and is topologically big.
Sur une courbe, qui remplit toute une aire plane
Über Peano-Kurven.
Whitney arcs and 1-critical arcs
A simple arc γ ⊂ ℝⁿ is called a Whitney arc if there exists a non-constant real function f on γ such that for every x ∈ γ; γ is 1-critical if there exists an f ∈ C¹(ℝⁿ) such that f’(x) = 0 for every x ∈ γ and f is not constant on γ. We show that the two notions are equivalent if γ is a quasiarc, but for general simple arcs the Whitney property is weaker. Our example also gives an arc γ in ℝ² each of whose subarcs is a monotone Whitney arc, but which is not a strictly monotone Whitney arc. This...
Необходимое и достаточное условие представимости конструктивных функций в виде суммы сингулярной и абсолютно непрерывной функции
Об одном аналоге понятия функций с ограниченным средним колебанием.