Jim writes [after my minor change, and addition, of notation]: << Say a function is locally unbounded at c if it is unbounded on every neighborhood of c. If a function f is differentiable everywhere, what can be said about the set of points LU[f'] at which the derivative f' is locally unbounded? Can LU[f'] be dense? Can it be all of R?
Jim defines a function [ditto]: << {x^2 Sin[1/x^2] for x not equal to 0, g(x) = { { 0 for x = 0
I'm going to guess that there exists a function h:R -> R differentiable everywhere with LU[f'] dense in R. 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 suspect this f *might* be differentiable everywhere with LU['h] dense in R. --Dan _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele