Let and then (where denotes greatest integer function)

MATHEMATICS · Single correct · CBSE · Mathematics · Continuity and Differentiability · Continuity and Differentiability

Problem

Let

f(x) = [x]

and

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

then (where

[.]

denotes greatest integer function) (1)

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

exists, but

g(x)

is not continuous at

x = 1

(2)

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

does not exist and

f(x)

is not continuous at

x = 1

(3)

x^{-1}

is continuous for all

x

(4)

\log x

is continuous for all

x