# $\dpi{100} f$ is defined in [ -5 , 5 ] as Option 1) $\dpi{100} f(x)\; is \; continuous\; at\; every\; x,except\; x=0$ Option 2) $\dpi{100} f(x)\; is \; discontinuous\; at\; every\; x,except\; x=0$ Option 3) $\dpi{100} f(x)\; is \; continuous\; everywhere$ Option 4) $\dpi{100} f(x)\; is \; discontinuous\; everywhere$

As we learnt in

Condition for discontinuity -

$1. \:L\neq R$

$\lim_{x\rightarrow a^{-}}\:f(x)=\lim_{x\rightarrow a^{+}}\:f(x)$

limit of function at x = a does not exist.

$2.\:L=R\neq V$

limit exist but not equal to  x = a

-

$f(x)=\left\{\begin{matrix} x& x\ is\ rational\\ -x& x\ is\ irrational \end{matrix}\right.$

ab $x=0$ both  are 0 so thas

$f\left (x \right ) is$ continuous at  $x=0$

Option 1)

$\dpi{100} f(x)\; is \; continuous\; at\; every\; x,except\; x=0$

Incorrect

Option 2)

$\dpi{100} f(x)\; is \; discontinuous\; at\; every\; x,except\; x=0$

Correct

Option 3)

$\dpi{100} f(x)\; is \; continuous\; everywhere$

Incorrect

Option 4)

$\dpi{100} f(x)\; is \; discontinuous\; everywhere$

Incorrect

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