Tuesday, 23 April 2013

real analysis - Nowhere continuous function for every equivalence class



Since our calculus lectures, we know that there are nowhere continuous functions (like the indicator function of the rationals). However, if we change this Dirichlet function on a set of measure zero, then we get a continuous function (the zero function). So my question ist





Does there exist a function f:RR such that every function which equals f almost everywhere (with respect to the Lebesgue measure) is nowhere continuous?
Is it possible to choose f borel-measurable?



Answer



See my answer to this question for the construction of an Fσ set MR such that 0<m(MI)<m(I) for every finite interval I, where m is the Lebesgue measure.



Let f be the indicator function of M. Clearly f is Borel-measurable. If g(x)=f(x) almost everywhere, then g1(0) and g1(1) are everywhere dense, whence g is nowhere continuous.


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