Are there any theories of integration out there that would make the function {-(2 Cos[1/x^2])/x + 2 x Sin[1/x^2] for x not equal to 0, f(x) = { { 0 for x = 0 integrable on the interval [-1,1]? Note that the function f(x), although discontinuous at x=0, is equal to the derivative of the function {x^2 Sin[1/x^2] for x not equal to 0, F(x) = { { 0 for x = 0 for ALL x (including x = 0). So, if we were to blindly apply the Fundamental Theorem of the Calculus without attending to the satisfaction of its hypotheses, we would conclude that the integral of f from -1 to 1 equals F(1) - F(-1). Proceeding less blindly, it seems safe to say that any theory of integration that assigned a value to the integral would have to assign it the value F(1) - F(-1). But the unbounded behavior of f(x) in the vicinity of x=0 makes things difficult. Indeed, the function f(x) is neither Riemann integrable nor Lebesgue integrable on [-1,1]. See http://jamespropp.org/142/FunctionsBehavingWorse.nb for a sketch of the graph. (Can nonstandard analysis help here? For some reason I have a feeling that the Stone-Cech compactification might be of use.) Another question arising from this example is: Can things be even worse? Say a function is locally unbounded at a if it is unbounded on every neighborhood of a. If a function F is differentiable everywhere, what can be said about the points at which the derivative is locally unbounded? Can it be dense? Can it be all of R? Jim