The class of measurable functions contains more functions than the class of continuous functions.The known result is that within the class of continuous functions,basic elementary functions such as linear function,exponential function,cosine function,and sine function can be characterized by a set of functional e-quations respectively.Using the absolute continuity of Lebesgue integrals,Lebesgue's differential theorem and Luzin's theorem,it can be proved that these characterizations still hold true within the class of measurable functions.
Real variable functionMeasurable functionAbsolute continuityIntegral of variable upper boundLuzin's theorem
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