Edit: I misread your other thread.
Okay so I just saw your other thread.
I think you want to show that
is Riemann integrabole on [0,1]
Here is one method let
is continuous on so it is uniformly continuous on
Since f is unifomly continous on [0,1] there exists a such that
Now let be any partition of [0,1] such that . Then by the extreme value theorem there are points are the max and min on for each j.
Since the mesh of P is less than delta we have so
So the upper and lower sums are