Define the function by .

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

Problem

Define the function

g: \mathbb{R} \to \mathbb{R}

g(x)=\lfloor x\rfloor+\lfloor -x\rfloor

. (i) Prove that

g(x)=-1

whenever

x

is not an integer. (ii) Find the value of

g(x)

when

x

is an integer. (iii) Sketch the graph of

g(x)

for

-3\le x\le 3

, clearly indicating all points of discontinuity.