KöMaL 2019 B5061

A function $f\colon \mathbb R \to \mathbb R$ is called area preserving if the area of the triangle formed by the points $\big(a,f(a)\big)$, $\big(b,f(b)\big)$ and $\big(c,f(c)\big)$ is equal to the area of the triangle formed by points $\big(a+x,f(a+x)\big)$, $\big(b+x,f(b+x)\big)$ and $\big(c+x,f(c+x)\big)$ for all $a<b<c$ and $x$. Which continuous functions $f$ are area preserving?