Define the function q(x)=⌊x+21⌋ for all real x.
a. (i) Prove that for any integer n, q(x)=n whenever n−21≤x<n+21.
b. (ii) Using part (i), list the values taken by q(x) for all x in the interval −3≤x<3.
c. (iii) Identify and list all points of discontinuity of q(x) in −3≤x≤3.
d. (iv) Sketch the graph of y=q(x) for −3≤x≤3, marking clearly the constant steps, open and closed endpoints, and the discontinuities.