Function of several real variables

In mathematical analysis, and applications in geometry, applied mathematics, engineering, natural sciences, and economics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The 'input' variables take real values, while the 'output', also called the 'value of the function', may be real or complex. However, the study of the complex valued functions may be easily reduced to the study of the real valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real valued functions will be considered in this article. In mathematical analysis, and applications in geometry, applied mathematics, engineering, natural sciences, and economics, a function of several real variables or real multivariate function is a function with more than one argument, with all arguments being real variables. This concept extends the idea of a function of a real variable to several variables. The 'input' variables take real values, while the 'output', also called the 'value of the function', may be real or complex. However, the study of the complex valued functions may be easily reduced to the study of the real valued functions, by considering the real and imaginary parts of the complex function; therefore, unless explicitly specified, only real valued functions will be considered in this article. The domain of a function of n variables is the subset of ℝn for which the function is defined. As usual, the domain of a function of several real variables is supposed to contain an open subset of ℝn. A real-valued function of n real variables is a function that takes as input n real numbers, commonly represented by the variables x1, x2, ..., xn, for producing another real number, the value of the function, commonly denoted f(x1, x2, ..., xn). For simplicity, in this article a real-valued function of several real variables will be simply called a function. To avoid any ambiguity, the other types of functions that may occur will be explicitly specified. Some functions are defined for all real values of the variables (one says that they are everywhere defined), but some other functions are defined only if the value of the variable are taken in a subset X of ℝn, the domain of the function, which is always supposed to contain an open subset of ℝn. In other words, a real-valued function of n real variables is a function such that its domain X is a subset of ℝn that contains an open set. An element of X being an n-tuple (x1, x2,..., xn) (usually delimited by parentheses), the general notation for denoting functions would be f((x1, x2,..., xn)). The common usage, much older than the general definition of functions between sets, is to not use double parentheses and to simply write f(x1, x2,..., xn). It is also common to abbreviate the n-tuple (x1, x2,..., xn) by using a notation similar to that for vectors, like boldface x, underline x, or overarrow →x. This article will use bold.