Computing the derivative of a function is difficult because, intuitively, the derivative depends on the local subtle changes of the function and is hard to compute from the approximation of the function. However, if some nice properties about the function is known (such as the differentiability of the derivative itself) then the derivative may be easy to compute. Formally, we prove that the derivative of a polynomial-time computable function is polynomial-time computable if and only if it has a polynomial modulus of continuity. Conversely, we can construct a function / in P C [0,1] such that its derivative exists everywhere but is not computable.
KeywordsPower Series Bounded Variation Modulus Function Computable Function Absolute Continuity
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