For every real

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

Problem

For every real

x

, let

h(x)=\lfloor 2x\rfloor-2\lfloor x\rfloor

. (i) Show that

0\le h(x)\le 1

for all real

x

. (ii) Prove that

h(x)

can take only the two values

0

and

1

, and no other real number. (iii) Find all real numbers

x

for which

h(x)=0

and all

x

for which

h(x)=1

. (iv) Hence write the range of

h

and comment on its discreteness.