Prove that the function f given by f(x) = x - [x] is increasing i… - Vidyadip

Question

Prove that the function f given by f(x) = x - [x] is increasing in (0, 1).

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Answer

f(x) = x - [x]

Let $\text{x}_1,\text{x}_2\in(0,1)$ such that x₁ < x_2.Then

[x₁] = [x₂] = 0 ....(1)

Now,

x₁ < x₂

⇒ x₁ - [x₁] < x₂ - [x₂] [From eq. (1)]

⇒ f(x₁) < f(x₂)

$\therefore$ x₁ < x₂

$\Rightarrow\text{f}(\text{x}_1)<\text{f}(\text{x}_2),\forall\ \text{x}_1,\text{x}_2\in(0,1)$

Hence, f(x) is increasing on (0, 1).

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