They are Riemann integrals. I see what you mean about not splitting it up into two cases. But I don't see why the minimum can't be larger than zero or why the maximum couldn't be less than zero :confused:
Suppose that the bounded function f:[a,b]-->R has the property that for each rational number x in the interval [a,b], f(x)=o for all x in [a,b]. Prove that
the lower integral of f from a to b is less than or equal to zero which is less than or equal to the upper integral of f from a to b...