# An infinitely differentiable function with compact support: Definition and properties

## TL;DR

This paper defines an infinitely differentiable function with compact support, explores its unique properties, and provides formulas for its derivatives, values at dyadic points, and related arithmetical characteristics.

## Contribution

It introduces a specific smooth function with compact support, detailing its properties, derivatives, and rationality of values at dyadic points, expanding understanding of such functions.

## Key findings

- Function is uniquely defined with compact support.
- Provides formulas for derivatives and values at dyadic points.
- Establishes arithmetical properties of the function's values.

## Abstract

This is the English translation of my old paper 'Definici\'on y estudio de una funci\'on indefinidamente diferenciable de soporte compacto', Rev. Real Acad. Ciencias 76 (1982) 21-38. In it a function (essentially Fabius function) is defined and given its main properties, including: unicity, interpretation as a probability, partition of unity with its translates, formulas for its $n$-th derivates, rationality of its values at dyadic points, formulas for the effective computation of these values, and some arithmetical properties of these values. Since I need it now for a reference, I have translated it.