Complexity theory of real functions
Complexity theory of real functions
Computable analysis: an introduction
Computable analysis: an introduction
Weakly Computable Real Numbers and Total Computable Real Functions
COCOON '01 Proceedings of the 7th Annual International Conference on Computing and Combinatorics
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The classical Jordan decomposition Theorem says that any real function of bounded variation can be expressed as a difference of two increasing functions. This paper explores the effective version of Jordan decomposition. We give a sufficient and necessary condition for those computable real functions of bounded variation which can be expressed as a difference of two computable increasing functions. Using this condition, we prove further that there is a computable real function which has even a computable modulus of absolute continuity (hence is of bounded variation) but it is not a difference of any two computable increasing functions. The polynomial time version of this result holds too and this gives a negative answer to an open question of Ko in [6].