$\"\"$

The function is $\"\"$ .

If $\"\"$ is a rational number then the function is $\"\"$ .

If $\"\"$ is a irrational number then the function is $\"\"$ .

$\"\"$ is never continuous because rational and irrational numbers are next to each other on the number line, $\"\"$ jumps between $\"\"$ and $\"\"$ .

The limit of the function does not exist.

Therefore the function $\"\"$ is never continuous.

$\"\"$

$\"\"$ does not exist.