Suppose I have a function $f$ defined as
$$f(y) = \{z \in [0, 1] \mid (ay-c) z \leq (ay - c)x\}$$
Where $y \in [0, 1], x \in [0, 1]$ and $ a, c \in \mathbb{R}$
Then claim:
$f(z) = 0$ if $(ay-c) > 0$
$f(z) = [0, 1]$ if $(ay-c) = 0$
$f(z) = 1$ if $(ay-c) < 0$
For the life of me I cannot reason out the $(ay-c) > 0$ and $(ay-c) < 0$ cases.
Can someone please clarify how they were able to arrive at this conclusion?
Hint: In those cases, the quantity is nonzero, so you can divide both sides of the defining inequality by it (in case it is positive, the inequality sign remains the same; in case it is negative, the inequality sign reverses). So the inequality will become either $$z\leq x$$ or $$z\geq x$$ Does that help?