Define by , where is the greatest integer less than or equal to and is the signum…

MATHEMATICS · Multi-part · AoPS · Mathematics · Special Functions · Special Functions

Problem

Define

f:\mathbb{R}\to\mathbb{R}

f(x)=\lfloor x \rfloor + \operatorname{sgn}(x)

, where

\lfloor x \rfloor

is the greatest integer less than or equal to

x

and

\operatorname{sgn}(x)

is the signum function. (i) State the definitions of the greatest integer function and the signum function. (ii) Evaluate

f(2.7)

f(-1.2)

and

f(0)

. (iii) If

x

varies in the interval

(-2,2)

, list all possible values that

f(x)

can take, giving reasons for your answer.