Define by , where denotes the greatest integer less than or equal to…

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

Problem

Define

q:\mathbb R\to\mathbb R

q(x)=[x]+1

, where

[x]

denotes the greatest integer less than or equal to

x

. (i) Identify all values of

x

in the interval

(-2,2)

at which

q

is not continuous. (ii) For any one of the discontinuity points found in (i), compute both

\displaystyle \lim_{x\to a^-}q(x)

and

\displaystyle \lim_{x\to a^+}q(x)

. (iii) Explain why the discontinuities of

q

(-2,2)

are of the same specific type. Name this type and state the characteristic feature that leads to your conclusion.