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

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There are infinitely many functions from S to T

There are infinitely many strictly increasing functions from S to T

The number of continuous functions from S to T is at most 120

Every continuous function from S to T is differentiable

answer is A, D.

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Detailed Solution

$S = (0, 1) \cup (1, 2) \cup (3, 4) T = {0, 1, 2, 3, 4}$

Number of functions :
Each element of S have 5 choice
Let n be the number of element in set S.
Number of function $= 5^{n}$
Here $n \to \infty$
$\Rightarrow$ Option (A) is correct.
Option (B) is incorrect
Number of strictly increasing functions $=^{5} C_{3} = 10$
(C) For continuous function
Each interval will have 5 choices.
$\Rightarrow$ Number of continuous functions
$= 5 \times 5 \times 5 = 125$
$\Rightarrow$ Option (C) is incorrect.
(D) Every continuous function is piecewise constant functions
$\Rightarrow$ Differentiable.
Option (D) is correct.