# Answering an open question in fuzzy metric spaces

## TL;DR

This paper proves that in stationary fuzzy metric spaces, the metric function and its slices are uniformly continuous, resolving an open problem and enhancing understanding of fuzzy metric space properties.

## Contribution

It provides a positive answer to an open question by establishing the uniform continuity of the fuzzy metric function and its slices in stationary fuzzy metric spaces.

## Key findings

- The function $M_y(x)$ is $ ext{R}$-uniformly continuous for all $y$ in $X$.
- The fuzzy metric function $M$ itself is $ ext{R}$-uniformly continuous.
- The paper resolves Problem 32 from GMM2012 affirmatively.

## Abstract

This paper answers affirmatively Problem 32 posted in \cite{GMM2012}, proving that, for every stationary fuzzy metric space $(X, M, *)$, the function $M_y(x):=M(x,y)$ defined therein is $\mathbb{R}$-uniformly continuous for all $y\in X$, and furthermore proves that the function $M$ is $\mathbb{R}$-uniformly continuous.