Real Analysis Exchange

Minimal Degrees of Genocchi-Peano Functions: Calculus Motivated Number Theoretical Estimates

Krzysztof Chris Ciesielski

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


A rational function of the form \(\frac{x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n}}{x_1^{\beta_1} + x_2^{\beta_2}+\cdots + x_n^{\beta_n}}\) is a Genocchi-Peano example, GPE, provided it is discontinuous, but its restriction to any hyperplane is continuous. We show that the minimal degree \(D(n)\) of a GPE of \(n\)-variables equals \(2\left\lfloor \frac{e^2}{e^2-1} n \right\rfloor+2i\) for some \(i\in\{0,1,2\}\). We also investigate the minimal degree \(D_b(n)\) of a bounded GPE of \(n\)-variables and note that \(D(n)\leq D_b(n)\leq n(n+1)\). Finding better bounds for the numbers \(D_b(n)\) remains an open problem.

Article information

Real Anal. Exchange, Volume 43, Number 2 (2018), 281-292.

First available in Project Euclid: 27 June 2018

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas
Secondary: 26B05: Continuity and differentiation questions

separate continuity hyperplane continuity smallest degree Genocchi-Peano examples


Ciesielski, Krzysztof Chris. Minimal Degrees of Genocchi-Peano Functions: Calculus Motivated Number Theoretical Estimates. Real Anal. Exchange 43 (2018), no. 2, 281--292. doi:10.14321/realanalexch.43.2.0281.

Export citation