I have recently stumbled across the Thomae's function, and the most interesting trait of this function are (obviously) its discontinuities. Still, I'm amazed by the converse: this function is actually continuous at irrationals.
My calculus background could be wider, and I know very little of $\mathbb{R}$'s properties, or set theory other than the naive version. I know that, by the definition of continuity of a function, there must be some interval around an irrational number where there are only irrational numbers, no matter how small. This would satisfy the definition: $$\lim_{x\to{x_0}} T(x) = T(x_0) = 0$$ But here's the point. Stating that there is an interval between rationals that only contains irrationals is something I can't quite rationalize (no pun intended), mostly because one proves easily that there's no such thing as "the next rational number".
I'm certain this relates to the density of irrationals in $\mathbb{R}$, but how does "deeper" mathematics (deeper as in, any theory concerning the problem at hand) handle this situation? Is there really an interval of irrationals between every rational, or am I missing something?
There is no interval $(a,b)\subset\mathbb R$ containing only irrational numbers. In fact, for any two real numbers $a$ and $b$ with $a<b$, there is a rational number $q$ such that $a<q<b$; in other words, the rationals are dense in $\mathbb R$.