Define the function for all real .

MATHEMATICS · Multi-part · ISC · Mathematics · Sets and Functions · Sets and Functions

Problem

Define the function

q(x)=\bigl\lfloor x+\dfrac12 \bigr\rfloor

for all real

x

. (i) Prove that for any integer

n

q(x)=n

whenever

n-\dfrac12 \le x < n+\dfrac12

. (ii) Using part (i), list the values taken by

q(x)

for all

x

in the interval

-3\le x<3

. (iii) Identify and list all points of discontinuity of

q(x)

-3\le x\le 3

. (iv) Sketch the graph of

y=q(x)

for

-3\le x\le 3

, marking clearly the constant steps, open and closed endpoints, and the discontinuities.