Let denote the greatest integer less than or equal to

MATHEMATICS · Multi-part · AP · Calculus AB · Limits and Continuity · Limits and Continuity

Problem

Let

\lfloor x\rfloor

denote the greatest integer less than or equal to

x

. Define

f(x)=\begin{cases} x-\lfloor x\rfloor+k, & x\notin\mathbb{Z},\\ 0, & x\in\mathbb{Z}, \end{cases}

where

k

is a real constant. (i) Calculate the left-hand limit and the right-hand limit of

f(x)

at an arbitrary integer

n\in\mathbb{Z}

in terms of

k

. (ii) Hence investigate whether there exists a real value of

k

for which

f

is continuous at every integer. Give full justification for your answer. (iii) State all points, if any, at which

f

is continuous when the value of

k

obtained in part (ii) is used.