ILL 1987 Problem 11

Let $S \subset [0, 1]$ be a set of 5 points with $\{0, 1\} \subset S$ . The graph of a real function $f : [0, 1] \to [0, 1]$ is continuous and increasing, and it is linear on every subinterval $I$ in $[0, 1]$ such that the endpoints but no interior points of $I$ are in $S$ .

We want to compute, using a computer, the extreme values of $g(x, t) = \frac{f(x+t)-f(x)}{ f(x)-f(x-t)}$ for $x - t, x + t \in [0, 1]$ . At how many points $(x, t)$ is it necessary to compute $g(x, t)$ with the computer?