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.

  • 1
    $\begingroup$ No this does not work. Check Dirichlet function. $\endgroup$ – Paramanand Singh Apr 22 at 5:56

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