Find out the values for which the function is continuous [duplicate]

calculus

Suppose $a=0$ and $(x_n) $ be any sequence such that $x_n \to 0 $

Then, $f(a) =0 $

If $f(x_n) =0 $ then done.

And in other case , $f(x_n)=x_n \to 0=f(0) $

Hence, $f$ is continuous at $0$

Suppose, $a\neq 0$ is rational.

Consider a sequence of irrational $(s_n) $ such that $s_n \to a$

Then, $f(s_n)=s_n \to a \neq 0=f(a) $

Suppose, $a\neq 0$ is irrational.

Consider a sequence of rational $(r_n) $ such that $r_n \to a$

Then, $f(r_n)\to 0 \neq a =f(a) $

Hence, the function $f$ is not continuous at any non zero point.

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