JEE Advanced 2023 · Paper 1, Mathematics question 1

(A) True. $S$ is infinite and $T$ has 4 elements, so there are infinitely many maps $S\to T$.

(B) False. A strictly increasing function is one-to-one. An infinite set cannot map one-to-one into the 4-element set $T$, so no such function exists.

(C) True. A continuous function into the discrete set $T$ is constant on each of the three intervals. That gives $4^3=64\le120$ functions.

(D) True. Each such function is locally constant, so its derivative exists and is 0 at every point of $S$.