The function $f(x)= \frac{1}{\sqrt{x}}$ has been demonstrated to be Riemann integrable in [0,1] not only from me, but also, in excellent fashion, by NonCommonAlg in the post I have reported...

... so for me the discussion is finished and, as Dante Alighieri in my signature, ...

...I don't like wasting time...

Kind regards

$\chi$ $\sigma$