# Riemann Integrable definition in partition distingued $\mathbb{Q}$

A function $$f:[a,b]\rightarrow \mathbb{R}$$ is Riemann Integrable on $$[a,b]$$ if
$$\exists \thinspace L \in \mathbb{R} : \forall \epsilon >0\thinspace \exists \thinspace \delta >0 :$$ if $$P$$ is any tagged partition of $$[a,b]$$ with $$\|P\|<\delta$$ then
$$\|S(f,P)-L\|<\epsilon$$ , where $$S(f,P)=\sum_{i=1}^{n}f(t_{i})(x_{i}-x_{i-1})$$ is Riemann sum, $$t_{i}\in[x_{i-1},x_{i}]$$ and $$\|P\|:=\max\{|x_i-i_{i-1}|1\leq i\leq n\}$$ Mi question is this definition is equal if i take $$t_{i} \in \mathbb{Q}$$ i don't know maybe is obvious for density i need a explain, thanks in advance.

• No this does not work. Check Dirichlet function. – Paramanand Singh Apr 22 at 5:56