JEE MAIN - Mathematics (2002 - No. 9)
$$f$$ is defined in $$\left[ { - 5,5} \right]$$ as
$$f\left( x \right) = x$$ if $$x$$ is rational
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ = - x$$ if $$x$$ is irrational. Then
$$f\left( x \right) = x$$ if $$x$$ is rational
$$\,\,\,\,\,\,\,\,\,\,\,\,\,$$ $$ = - x$$ if $$x$$ is irrational. Then
$$f(x)$$ is continuous at every x, except $$x = 0$$
$$f(x)$$ is discontinuous at every $$x,$$ except $$x = 0$$
$$f(x)$$ is continuous everywhere
$$f(x)$$ is discontinuous everywhere
Explanation
Let a is a rational number other than $$0,$$ in
$$\left[ { - 5,5} \right],$$ then
$$f\left( a \right) = a$$ and $$\mathop {\lim }\limits_{x \to a} \,\,f\left( x \right) = - a$$
[ 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
$$\,f\left( a \right) = - a$$ and $$\mathop {\lim }\limits_{x \to a} \,f\left( x \right) = a$$
$$\therefore$$ $$f(x)$$ is not continuous at any irrational number clearly
$$\mathop {\lim }\limits_{x \to 0} f\left( x \right) = f\left( 0 \right) = 0$$
$$\therefore$$ $$f(x)$$ is continuous at $$x=0$$
$$\left[ { - 5,5} \right],$$ then
$$f\left( a \right) = a$$ and $$\mathop {\lim }\limits_{x \to a} \,\,f\left( x \right) = - a$$
[ 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
$$\,f\left( a \right) = - a$$ and $$\mathop {\lim }\limits_{x \to a} \,f\left( x \right) = a$$
$$\therefore$$ $$f(x)$$ is not continuous at any irrational number clearly
$$\mathop {\lim }\limits_{x \to 0} f\left( x \right) = f\left( 0 \right) = 0$$
$$\therefore$$ $$f(x)$$ is continuous at $$x=0$$
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