From: israel@math.ubc.ca (Robert Israel) Newsgroups: sci.math Subject: Re: continuous a.e. Date: 9 Mar 1998 09:56:05 GMT In article , G. A. Edgar wrote: >Perhaps it would be clearer if I said >[c] > There is a set B such that A-B has measure zero and f is continuous on > B relative to B. >to distinguish it from >[b] > There is a set B such that A-B has measure zero and f is continuous at > every point of B (relative to A). [b] is the usual meaning, I think. Here's a Lebesgue-measurable function on A=R that does not satisfy [c]. Let {(a_n, b_n)} be an enumeration of the open intervals with rational endpoints, and for each n let C_n be a nowhere-dense closed set of positive measure disjoint from all previous C_n. Define f(x) = n if x in C_n, f(x) = 0 if x is in no C_n. Suppose B is any set with m(R-B)=0. I claim f is nowhere continuous on B (relative to B). Let b in B with f(b) = k (thus either b in C_k or b not in any C_n). Given any delta > 0, take an interval (a_n, b_n) for n > k contained in (b-delta, b+delta). Since m(C_n-B)=0, B must contain points of C_n, i.e. there is y in B with |y-b| < delta and f(y) = n >= 1 + f(b). Robert Israel israel@math.ubc.ca Department of Mathematics (604) 822-3629 University of British Columbia fax 822-6074 Vancouver, BC, Canada V6T 1Z2