Question

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

Answer 1

$$\begin{gathered} f\left(x\right)=x-\left[x\right] \\ \text{Let}{x}_1,x_2\in \left(0,1\right)\text{such that}{x}_1<x_2.\mathrm{Then}, \\ \left[x_1\right]\text{=}\left[x_2\right]\text{= 0 ...(1)} \\ \text{Now,} \\ {x}_1<x_2 \\ \Rightarrow x_1-\left[x_1\right]<x_2-\left[x_2\right]\left[\text{From eq. (1)}\right] \\ \Rightarrow f\left(x_1\right)<f\left(x_2\right) \\ \text{∴}{x}_1<x_2 \\ \Rightarrow f\left(x_1\right)<f\left(x_2\right),\forall x_1,x_2\in \left(0,1\right) \\ \text{So,}f\left(x\right)\text{is increasing on}\left(0,1\right)\text{.} \end{gathered}$$