Dan Asimov writes:
Using this function g and an enumeration {q_n}, n = 1,2,3,... of the rationals, define
h:R -> R via h(x) := Sum_{n=1,2,3,...} g(x-q_n)/2^n
I like Dan's idea of translating the graph of x^2 sin 1/x^2 horizontally by a dense set of displacements and summing (with coefficients falling off quickly enough to encourage summability). But I'm worried that if an adversary picks the enumeration of the rationals, the sum won't converge everywhere (because x^2 sin 1/x^2 can be pretty big when x is large). Fortunately this issue can be ducked by using the bounded function x^2 sin 1/x^2 exp -x^2. Andy Latto asks:
Can you provide a hint of the proof that it is not Lebesgue integrable?
I believe that the total signed area between the graph of f(x) and the x-axis is not absolutely summable. I.e., the integral of max(f(x),0) on [-1,1] is infinite. Fred Lunnon writes:
Only somebody with Nota Bene installed is going to be able to view this!
Actually ".nb" here means "Mathematica notebook". I'll make everyone's life easier by putting a PDF on the web (though I should mention that there's now free software for viewing Mathematica notebooks): See http://jamespropp.org/142/FunctionsBehavingWorse.pdf . Jim