Strongly ℚ-Differentiable Functions

Abstract

A real function is called strongly $\mathbb{Q}$-dif\-fer\-en\-ti\-able if, for every real number $ h \,$, the limit of the ratio $ \left( f(x+rh) - f(x) \right) / r $ exists whenever $x$ tends to any fixed real number and $r$ tends to zero through the positive rationals. After examining the dependence of strong $\mathbb{Q}$-derivatives on their parameters, we prove that every strongly $\mathbb{Q}$-dif\-fer\-en\-ti\-able function can be represented as the sum of an additive mapping and a continuously dif\-fer\-en\-ti\-able function.