Darboux ’ s Theorem
نویسندگان
چکیده
We adopt the following convention: x, y are real numbers, i, j, k are natural numbers, and p, q are finite sequences of elements of R. The following propositions are true: (1) Let A be a closed-interval subset of R and D be an element of divsA. If vol(A) 6= 0, then there exists i such that i ∈ domD and vol(divset(D, i)) > 0. (2) Let A be a closed-interval subset of R, D be an element of divsA, and given x. If x ∈ A, then there exists j such that j ∈ domD and x ∈ divset(D, j). (3) Let A be a closed-interval subset of R and D1, D2 be elements of divsA. Then there exists an element D of divsA such that D1 ¬ D and D2 ¬ D and rngD = rngD1 ∪ rngD2.
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