Abstract
It is well known that any real‐valued function f continuous on the real line and satisfying the functional relationship f(a + b)= f(a) + f(b)for all real aand bmust be of the form f(x) = cxwhere cis a constant. Further, concepts such as the axiom of choice, well ordering, and Hamel bases can be employed to prove that there exists a function f discontinuous everywhere on the real line such that f(a + b) =f(a) + f(b)for all real aand b.
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