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An elementary example which any grad can think about. Hope it will require 10 mins for most of them to figure out (or recall) the proof.: The set of the discontinuous points of a non-decreasing function. Or, making it more geometric, (under suitable assumptions) the set of the radius $r$ such that a given Borel measure charges the $r$-sphere (with suitable assumptions)r$-sphere. Or, based on the first result, the set of non-differentiable points of a convex function on the real line. |
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An elementary example which any grad can think about. Hope it will require 10 mins for most of them to figure out (or recall) the proof. The number set of the discontinuous points of a non-decreasing functionon the real line. Or, making it more geometric, the number set of the radius $r$ such that a given Borel measure charges the $r$-sphere (with a fixed centre, say $0$).suitable assumptions). |
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An elementary example which any grad can think about. Hope it will require 10 mins for most of them to figure out (or recall) the proof. The number of discontinuous points of a non-decreasing function on the real line. Or, making it more geometric, the number of the radius $r$ such that a given Borel measure charges the $r$-sphere (with a fixed centre, say $0$). |
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