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

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

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