Consider function $f $ defined on an interval $[-1,1] $. Evaluate whether $f $ is continuous, Riemann integrable and/or Newton integrable on the given interval.
$f(x) = \left\{\begin{matrix}x ,& x\in \mathbb{Q} \\0 ,& \text{otherwise} \\\end{matrix}\right. $
- Continuity
I think that $f $ is almost continuous on the given interval since the points of discontinuity are points from $\mathbb{Q}$, which is a null set, thus by Lebesgue's theorem (A bounded function is Riemann integrable iff it's almost continuous) it should also indidicate a possible Riemann-integrability.
- Newton-integrability
The function $f$ does not have the Darboux's property, thus $f$ does not have a primitive function and it is not Newton-integrable.
(using the theorem that says "If $f$ has a primitive function on interval $I$, then $f$ has the darboux property on interval $I$.")
- Riemann-integrability
The truth is, I still struggle with understand why Riemann's function (Thomae function) is Riemann-integrable whereas Dirichlet's function isn't. I know the proof using the upper and lower sums that show that Dirichlet's function isn't Riemann integrable ... by why can we use the argument that "$\mathbb{Q}$ form a null set and according to Lebesgue's theorem, the function is Riemann integrable$ for Thomae function but it doesn't work with Dirichlet's function?
That's also why I'm struggling to figure out whether I can or cannot use this argument for my function $f$.
