Let if 0 < x < 1, otherwise. Let be an enumeration of rationals, and define . Show that g is discontinuous at every point and unbounded on every interval.
Show that if f is a non-negative integrable function, for every there exists such that and
Let if 0 < x < 1, otherwise. Let be an enumeration of rationals, and define . Show that g is discontinuous at every point and unbounded on every interval.
Show that if f is a non-negative integrable function, for every there exists such that and
I have no clue about the first one
For question #2:
I'm not sure what M is? Is that the domain of the function? Assuming it is
Let = {x \in M: f(x) > L }
Notice that and thus
Since f is integrable, the numerator is bounded and it follows that as L tends to infinity, tends to 0.
I'll leave it up to you to show that as and so we can find an L large enough such that and so the set we are looking for is E = M \
Note, this used the fact that we can break the integral up into disjoint sets.