#### Question

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

#### Solution

\[f\left( x \right) = x - \left[ x \right]\]

\[\text { Let } x_1 , x_2 \in \left( 0, 1 \right) \text { such that } x_1 < x_2 . \text { Then }, \]

\[\left[ x_1 \right]=\left[ x_2 \right]= 0 ...(1)\]

\[\text { Now,}\]

\[ x_1 < x_2 \]

\[ \Rightarrow x_1 - \left[ x_1 \right] < x_2 - \left[ x_2 \right] \left[ \text { From eq }. (1) \right]\]

\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right)\]

\[\therefore x_1 < x_2 \]

\[ \Rightarrow f\left( x_1 \right) < f\left( x_2 \right), \forall x_1 , x_2 \in \left( 0, 1 \right)\]

\[\text { So},f\left( x \right) \text { is increasing on }\left( 0, 1 \right).\]

Is there an error in this question or solution?

Solution Prove that the Function F Given by F(X) = X − [X] is Increasing in (0, 1) ? Concept: Increasing and Decreasing Functions.