This function appears without any reference in the book The Irrationals :
$$\lim_{m\to\infty}\lim_{n\to\infty} \cos^{2n}(m!\pi x)=\left\{ \begin{array}{lr} 1 & : x \in \mathbb{Q}\\ 0 & : x \notin \mathbb{Q} \end{array} \right.$$
I found the name of the function to be Dirichlet's function and it got me thinking. Is this function enough to prove constants like $\gamma$ irrational?
I am a beginner when it comes to irrationality proofs, but I think this function requires one to know the entire decimal expansion of the number, $x$, in question because $m!$ goes to infinity and a decimal approximation for $x$ would eventually make the limit go to $1$...?
Right now it is unknown if $\gamma$ is irrational or not. As long there is no proof for this result, it is improbable that one can comment if a certain technique might be helpful for this....