We usually use Riemann sums to define the integral. This require the existence of the function over [0,1] and the continuity of it over ]0,1[.
Show that and I think it will be sufficient.
First part is easily checked.
For second part, show that there is a discontinuity at 0. It follows that the set is open (since infinity is not included) and it is therefore continuous over ]0,1[.