Let and . Then which of the following is not true ( represents the greatest integer…

MATHEMATICS · Single correct · AoPS · Mathematics · Special Functions · Special Functions

Problem

Let

f(x)=[x]

and

g(x)=\begin{cases}0, & x\in\mathbb{Z}\\ x^2, & x\in\mathbb{R}-\mathbb{Z}\end{cases}

. Then which of the following is not true (

[\cdot]

represents the greatest integer function)? (a)

\lim_{x\rightarrow 1} g(x)

exists but

g(x)

is not continuous at

x=1

. (b)

\lim_{x\rightarrow 1} f(x)

does not exist and

f(x)

is not continuous at

x=1

. (c)

g\circ f

is a discontinuous function. (d)

f\circ g

is a discontinuous function.