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Question
Prove that the function f given by f(x) = x − [x] is increasing in (0, 1) ?
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Solution
\[f\left( x \right) = x - \left[ x \right]\]
\[\text { Let } x_1 , x_2 \in \left( 0, 1 \right) \text { such that } x_1 < x_2 . \text { Then }, \]
\[\left[ x_1 \right]=\left[ x_2 \right]= 0 ...(1)\]
\[\text { Now,}\]
\[ x_1 < x_2 \]
\[ \Rightarrow x_1 - \left[ x_1 \right] < x_2 - \left[ x_2 \right] \left[ \text { From eq }. (1) \right]\]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]
\[\therefore x_1 < x_2 \]
\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, 1 \right)\]
\[\text { So},f\left( x \right) \text { is increasing on }\left( 0, 1 \right).\]
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