Non-symmetrically $t$-affine functions revisited

In 2014, Michal Lewicki and Andrzej Olbryś proved that if a real valued function $f$ defined on the real line satisfies the conditional functional equation $$f(tx+(1-t)y)=tf(x)+(1-t)f(y),x\le y,$$ 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 Olbryś for any subinterval of $\mathbb{R}$.