$\"\"$

The dirichlet function is $\"\"$ .

$\"\"$ 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 $\"\"$ .

$\"\"$ does not exist.

Therefore, the function is not continuous at any real number.

$\"\"$

The function is not continuous at any real number. \ \