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 $\displaystyle\lim_{x \to a} f(x) = - a$
[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 $\displaystyle\lim_{x \to a} f(x) = a$
$\therefore \, \, f(x)$ is not continuous at any irrational number clearly
$\displaystyle\lim_{x \to 0} f(x) = f(0) = 0$
$ \therefore \, f(x)$ is continuous at x = 0