: Combination of Greatest–Integer and Signum Functions

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

Problem

: Combination of Greatest–Integer and Signum Functions (a) Let

p(x)=\left\lfloor\dfrac{x}{2}\right\rfloor+\operatorname{sgn}(x)+1

for all real

x

. Answer the following questions: (b) (i) Name the family of function represented by each symbol that appears in the definition of

p(x)

. (c) (ii) Rewrite

p(x)

as an explicit piecewise rule for

-3\le x<3

, labelling the family of each piece. (d) (iii) Draw a neat sketch of

y=p(x)

for the interval

-3\le x<3

. (e) (iv) Using the sketch obtained in part (iii), list all values of

x

[-3,3)

for which

p

is constant on some open interval containing

x

. Give reasons for your answer.