element of several different Hilbert spaces. Complex-valued function of a real variable A complex-valued function of a real variable may be defined by relaxing, in the definition of the real-valued functions, the restriction of the codomain to the real numbers, and allowing complex values. If is such a complex valued function, it may be decomposed as = + , where and are real-valued functions. In other words, the study of the complex valued functions reduces easily to the study of the pairs of real valued functions. Cardinality of sets of functions of a real variable The cardinality of the set of real-valued functions of a real variable, , is , which is strictly larger than the cardinality of the continuum (i.e., set of all real numbers). This fact is easily verified by cardinal arithmetic: Furthermore, if is a set such that , then the cardinality of the set is also , since However, the set of continuous functions has a strictly smaller cardinality, the cardinality of the continuum, . This follows from the fact that a continuous function is completely determined by its value on a dense subset of its domain. Thus, the cardinality of the set of continuous real-valued functions on the reals is no greater than the cardinality of the set of real-valued functions of a rational variable. By cardinal arithmetic: On the other hand, since there is a clear bijection between and the set of constant functions , which forms a subset of , must also hold. Hence, . See also Real analysis Function of several real variables Complex analysis Function of several complex variables References External links Multivariable Calculus L. A. Talman (2007) Differentiability for Multivariable Functions Mathematical analysis Real numbers Multivariable calculus