I have a particular problem and its about Darboux sums and integrable functions. Here is a statement that is False, but want to demonstrate with example why its false. Here is the statement:
Let P be a partition, defined by, P ={1,1.3,2} be a partition of the closed interval [1,2]. Let f(x) = 0 for the range x<1.5, and let f(x) = 4 for x>= 1.5 . The the upper sum U(f,P) = 2.8 and the lower sum L(f,P) = 0. Therefore, function f is NOT integrable on [1,2] by the Darboux definition of the definite integral.
SO this statement is false. This function is integrable over this interval of [1,2].
My undestanding of the Darboux definition for integrable is that of, IF one can find a particular partition of a given interval such that L(f,P) = U(f,P), then the function is integrable over that interval by the Darboux definition.
I now want to find a partition of [1,2] such that, L(f,P) = U(f,P).
But I am having difficulty figuring out the partition, choosing the numbers for this to work.
Hope someone can figure it out.