Prove that the function `f` given by `f(x)=x-[x]` us ubcreasubg ub `(0,1)dot`

Explanation

Here, `f(x) = x-[x]`

We know, `x = [x] +{x}`, where `[x]` is the greatest integer function of `x` and `{x}` is the fraction of `x`.

`:. x - [x] = {x}`.

`:. f(x) = {x}`

Let `x_1 = 0.23` and `x_2 = 0.25`

Then, `f(x_2) gt f(x_1)`

It means, if `x_2 gt x_1` , then, `f(x_2) gt f(x_1)` for `x in (0,1).`

Therefore, `f(x)` is an increasing function in `(0,1)`.