The simplest possible sampling of a function in the region [0,1] for the purpose of finding the area under the curve, for instance, could be just take the left offsets of the bars. That is, you have divided your [0,1] into n vertical bars and sample f(left border of every bar). If we take only 2 points, they will be 0 and .5. This has unsatisfactory shit to the left. 

To fix it, you decide to sample in the centres of the bars. That is, you have the same 2 sample points, you take them in .25 and .75 -- right in the middles of your two bars. Looking at it, you immediately realize that it is not better -- you sample too closely to the 0 and 1 but leave a big gap in between .25 and .75. You realize that fair sampling would be to have 3 subintervals instead of two and you divide [0,1] into n+1 sections by exactly n points. Now, two points will sample at .33 and .66 -- equidistantly both from the interval borders as well as from each other. How do you call this kind of uniform sampling?

Carry me over to the *math* pleas if you think that deterministic sampling that I address here is not suitable in probability/statistic area.