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

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Solution

$f (x) = x - [x]$
for $x \in (0, 1) \Rightarrow [x] = 0$

$∴ f (x) = x$
$f^{'} (x) = 1 > 0$
Hence, $f (x)$ is increasing for $x \in (0, 1)$ .

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