Function of a real variable in the context of Critical point (mathematics)

Mathematical optimization (alternatively spelled optimisation) or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfields: discrete optimization and continuous optimization. Optimization problems arise in all quantitative disciplines from computer science and engineering to operations research and economics, and the development of solution methods has been of interest in mathematics for centuries.

In the more general approach, an optimization problem consists of maximizing or minimizing a real function by systematically choosing input values from within an allowed set and computing the value of the function. The generalization of optimization theory and techniques to other formulations constitutes a large area of applied mathematics.

In mathematics, a real-valued function is a function whose values are real numbers. In other words, it is a function that assigns a real number to each member of its domain.

Real-valued functions of a real variable (commonly called real functions) and real-valued functions of several real variables are the main object of study of calculus and, more generally, real analysis. In particular, many function spaces consist of real-valued functions.

A triangular wave or triangle wave is a non-sinusoidal waveform named for its triangular shape. It is a periodic, piecewise linear, continuous real function.

Like a square wave, the triangle wave contains only odd harmonics. However, the higher harmonics roll off much faster than in a square wave (proportional to the inverse square of the harmonic number as opposed to just the inverse).

In mathematical analysis and its applications, 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 ⁠${\displaystyle \mathbb {R} ^{n}}$⁠ for which the function is defined. As usual, the domain of a function of several real variables is supposed to contain a nonempty open subset of ⁠${\displaystyle \mathbb {R} ^{n}}$⁠.

In mathematics, an elementary function is a function of a single variable (real or complex) that is typically encountered by beginners. The basic elementary functions are polynomial functions, rational functions, the trigonometric functions, the exponential and logarithm functions, the n-th root, and the inverse trigonometric functions, as well as those functions obtained by addition, multiplication, division, and composition of these. Some functions which are encountered by beginners are not elementary, such as the absolute value function and piecewise-defined functions. More generally, in modern mathematics, elementary functions comprise the set of functions previously enumerated, all algebraic functions (not often encountered by beginners), and all functions obtained by roots of a polynomial whose coefficients are elementary.

This list of elementary functions was originally set forth by Joseph Liouville in 1833. A key property is that all elementary functions have derivatives of any order, which are also elementary, and can be algorithmically computed by applying the differentiation rules (or the rules for implicit differentiation in the case of roots). The Taylor series of an elementary function converges in a neighborhood of every point of its domain. More generally, they are global analytic functions, defined (possibly with multiple values, such as the elementary function ${\displaystyle {\sqrt {z}}}$ or ${\displaystyle \log z}$) for every complex argument, except at isolated points. In contrast, antiderivatives of elementary functions need not be elementary and is difficult to decide whether a specific elementary function has an elementary antiderivative.