$\"\"$

\

The function is $\"\"$ .

\

Find $\"\"$ .

\

$\"\"$ is does not exist, because there are infinite number of rational and irrational numbers near to $\"\"$ .

\

The function value of these infinite numbers are not approaching a unique value.

\

If $\"\"$ is rational the function value is $\"\"$ .

\

If $\"\"$ is irrational the function value is $\"\"$ .

\

Therefore, $\"\"$ does not exist.

\

$\"\"$

\

The function is $\"\"$ .

\

Find $\"\"$ .

\

$\"\"$ definition :

\

$\"\"$ means that for every $\"\"$ , there exists a $\"\"$ , such that for every $\"\"$ , the expression $\"\"$ implies $\"\"$ .

\

Consider $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

If $\"\"$ is rational, then $\"\"$ .

\

$\"\"$

\

$\"\"$ .

\

If $\"\"$ is irrational, then $\"\"$ .

\

$\"\"$ .

\

let $\"\"$ .

\

$\"\"$ .

\

\

Observe the relationship between two absolute values $\"\"$ and $\"\"$ .

\

$\"\"$

\

$\"\"$

\

Therefore $\"\"$ .

\

As $\"\"$ tends to $\"\"$ , $\"\"$ approaches to $\"\"$ .

\

$\"\"$ .

\

$\"\"$

\

$\"\"$ does not exist.

\

$\"\"$ .