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 $x_1 < x_{2.}$
Then
$[x_1] = [x_2] = 0 ....(1)$
Now,
$x_1 < x_2$
$\Rightarrow x_1 - [x_1] < x_2 - [x_2] [$From eq. $(1)]$
$\Rightarrow f(x_1) < f(x_2)$
$\therefore x_1 < x_2$
$\Rightarrow\text{f}(\text{x}_1)<\text{f}(\text{x}_2),\forall\ \text{x}_1,\text{x}_2\in(0,1)$
Hence, $f(x)$ is increasing on $(0, 1).$

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