JEE Advanced 2023 Paper 1 Q01 Mathematics Sets, Relations & Functions Functions & Composition Medium

JEE Advanced 2023 Paper 1 · Q01 · Functions & Composition

Let $S = (0,1) \cup (1,2) \cup (3,4)$ and $T = \{0, 1, 2, 3\}$. Then which of the following statements is(are) true?

  1. A. There are infinitely many functions from $S$ to $T$
  2. B. There are infinitely many strictly increasing functions from $S$ to $T$
  3. C. The number of continuous functions from $S$ to $T$ is at most $120$
  4. D. Every continuous function from $S$ to $T$ is differentiable
JEE Advanced multi-correct — pick every correct option, then check.
Reveal answer + step-by-step solution

Correct answer:A, C, D

Solution

Domain $S$ is infinite while codomain $T = \{0,1,2,3\}$ has $4$ elements, so option (A) is correct (infinitely many arbitrary functions). Strictly increasing functions from $S$ to a finite set are NOT possible because $S$ contains intervals, so (B) is false. A continuous function on a disjoint union of intervals must be constant on each component; with $3$ components and $4$ choices we get $4 \times 4 \times 4 = 64 \le 120$, so (C) is correct. Each component is an open interval and the function is locally constant, hence differentiable — so (D) is correct.

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