Mathematics

Problem

Let

f:\mathbb{R}\to\mathbb{R}

be defined by

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

where

k

is a real constant. a. Part 1: (a) Determine all real values of

k

for which

f

is continuous at

x=0

. b. Part 2: (b) Investigate whether there exists any integer

n\neq0

such that

f

can be made continuous at

x=n

by a suitable choice of

k

. Provide complete justification for your answer.