Question

f is defined in [-5, 5] as
f (x) = x if x is rational
= - x if x is irrational. Then

• A. f (x) is continuous at every x, except x = 0
• B. f (x) is discontinuous at every x, except x = 0
• C. f (x) is continuous everywhere
• D. f(x) is discontinuous everywhere

Answer 1

Answer: (B)

Let a is a rational number other than 0, in [-5, 5], then
$f(a)=a$ and $\underset{x\to a}{\lim }f(x)=-a$
[As in the immediate neighbourhood of a rational number, we find irrational numbers]
$\therefore f(x)$ is not continuous at any rational number If a is irrational number, then
$f(a)=-a$ and $\underset{x\to a}{\lim }f(x)=a$
$\therefore f(x)$ is not continuous at any irrational number clearly
$\underset{x\to 0}{\lim }f(x)=f(0)=0$
$\therefore f(x)$ is continuous at x = 0