Mathematics

Problem

Let

q:\mathbb{R}\to\mathbb{R}

be given by

q(x)=\lfloor x\rfloor

, where

\lfloor x\rfloor

denotes the greatest integer less than or equal to

x

. a. (i) Evaluate the left-hand limit and the right-hand limit of

q(x)

x=2

. b. (ii) Determine whether

q

is continuous from the right at

x=2

and whether it is continuous from the left at the same point. c. (iii) Using your results, classify the type of discontinuity that

q

exhibits at each integer

n\in\mathbb{Z}