SCHWEITZER 1963 Problem 9

Let $f(t)$ be a continuous function on the interval $0 \leq t \leq 1$ , and define the two sets of points $$A_t=\{(t,0): t\in[0,1]\} , B_t=\{(f(t),1): t\in [0,1]\}.$$ Show that the union of all segments $\overline{A_tB_t}$ is Lebesgue-measurable, and find the minimum of its measure with respect to all functions $f$ . [A. Csaszar]