JEE Advanced · 2023 · 04 Jun 2023 · Paper 1 · Relations And Functions XII

Option (A): Each element of S has 4 choices in T. Since S has infinitely many elements, the number of functions is $4^n$ where $n\to \infty$, so there are infinitely many functions. (A) is correct.

Option (B): Strictly increasing functions are limited by the discrete nature of T and the disconnected intervals of S. There cannot be infinitely many such functions. (B) is incorrect.

Option (C): For a continuous function from S to T, each of the three disjoint intervals $(0,1)$, $(1,2)$, and $(3,4)$ must map to a constant value in T (since T is discrete). Each interval has 4 choices, giving $4\times 4\times 4=64$ continuous functions, which is at most 120. (C) is correct.

Option (D): Every continuous function from S to T is piecewise constant (constant on each connected component), hence differentiable everywhere in S (derivative is zero). (D) is correct.

Answer: (A), (C), (D)