## Real Analysis Exchange

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

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

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 27 June 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.rae/1530064961

**Digital Object Identifier**

doi:10.14321/realanalexch.43.2.0281

**Mathematical Reviews number (MathSciNet)**

MR3942578

**Zentralblatt MATH identifier**

06924889

**Subjects**

Primary: 11A25: Arithmetic functions; related numbers; inversion formulas

Secondary: 26B05: Continuity and differentiation questions

**Keywords**

separate continuity hyperplane continuity smallest degree Genocchi-Peano examples

#### Citation

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. https://projecteuclid.org/euclid.rae/1530064961