 in mathematical analysis, and applications in geometry, applied mathematics, engineering, and natural sciences, a function of a real variable is a function whose domain is the real numbers â, more specifically the subset of â for which the function is defined. the "output", also called the "value of the function", could be anything: simple examples include a single real number, or a vector of real numbers (the function is "vector valued"). vector-valued functions of a single real variable occur widely in applied mathematics and physics, particularly in classical mechanics of particles, as well as phase paths of dynamical systems. but we could also have a matrix of real numbers as the output (the function is "matrix valued"), and so on. the "output" could also be other number fields, such as complex numbers, quaternions, or even more exotic hypercomplex numbers. ==general definition == thumb|a function of a real variable a real-valued function of a real variable is a function that takes as input a real number, commonly represented by the variable x, for producing another real number, the value of the function, commonly denoted f(x). for simplicity, in this article a real-valued function of a real variable 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 is taken in a subset x of â, the domain of the function, which is always supposed to contain an open subset of â. in other words, a real-valued function of a real variable is a function :f: x \rightarrow \mathbb{r} such that its domain x is a subset of â that contains an open set. a simple example of a function in one variable could be: : v : x \rightarrow \mathbb{r} : x = \{ x \in \mathbb{r} \,:\, 0 \leq x\} : f(x) = \sqrt{x} which is the square root ofx. ===image=== the image of a function f(x) is the set of all values of  when the variable x runs in the whole domain of . for a continuous (see below for a definition) real-valued function with a connected domain, the image is either an interval or a single value. in the latter case, the function is a constant function. the preimage of a given real number y is the set of the solutions of the equation . === domain === the domain of a function o
Mathematical_analysis Multivariable_calculus Real_numbers