# A note on $h$-convex functions

## TL;DR

This paper explores the properties of $h$-convex functions, including their continuity, geometric interpretation, and conditions under which $h$-midconvexity implies $h$-convexity, along with derivative characterizations.

## Contribution

It introduces the concept of $h$-convex curves and provides new insights into the continuity and geometric aspects of $h$-convex functions.

## Key findings

- $h$-convex functions are $h$-continuous
- $h$-midconvexity is equivalent to $h$-convexity for $h$-continuous functions
- Derivative characterizations of $h$-convexity are discussed

## Abstract

In this work, we discuss the continuity of $h$-convex functions by introducing the concepts of $h$-convex curves ($h$-cord). Geometric interpretation of $h$-convexity is given. The fact that for a $h$-continuous function $f$, is being $h$-convex if and only if is $h$-midconvex is proved. Generally, we prove that if $f$ is $h$-convex then $f$ is $h$-continuous. A discussion regarding derivative characterization of $h$-convexity is also proposed.