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Different definitions exist depending on the specific field of application. Traditionally, an '''additive function''' is a function that preserves the addition operation:
:''f''(''x'' + ''y'') = ''f''(''x'') + ''f''(''y'')
for any two elements ''x'' and ''y'' in the ___domain. An example of an additive function would include the total-deriviate operator d; that is to say d(x + y) = d(x) + d(y).
In [[number theory]], an '''additive function''' is an [[arithmetic function]] ''f''(''n'') of the positive [[integer]] ''n'' such that whenever ''a'' and ''b'' are [[coprime]], the function of the product is the sum of the functions:
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