Non-symmetrically $t$-affine functions revisited

TL;DR

This paper revisits the concept of non-symmetrically $t$-affine functions, extending previous results by showing local $t$-affinity and generalizing the main theorem to subintervals of the real line.

Contribution

It generalizes Lewicki and Olbryś's 2014 theorem to any subinterval of the real line, establishing local and interval $t$-affinity for such functions.

Findings

Conditional equation implies local $t$-affinity.

$t$-affinity holds on open intervals.

Main result extends to all subintervals of $ eal$.

Abstract

In 2014, Michal Lewicki and Andrzej Olbry\'s proved that if a real valued function $f$ defined on the real line satisfies the conditional functional equation \[ f(tx + (1-t)y) = t f(x) + (1-t) f(y),\qquad x\leq y, \] called non-symmetrically $t$-affine, then it is $t$-affine. That is, they concluded that $f$ must fulfill the above equality without any restriction on $x$ and $y$. In the current study, first we show that the above conditional equation implies that the function in question is locally $t$-affine. Then we derive $t$-affinity on open intervals. Finally, we formulate our main result, which generalizes the theorem of Lewicki and Olbry\'s for any subinterval of $\mathbb{R}$.