Question 135
Determine the continuous functions $f : \mathbb{R} \to \mathbb{R}$ having the property that, for all $x, y \in \mathbb{R}$, there exist $t \in (0, 1)$ such that
$$
f((1-t)x + ty) = (1-t)f(x) + tf(y).
$$
$$
f((1-t)x + ty) = (1-t)f(x) + tf(y).
$$