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Any continuous function has a fixed point, whereis the unit -ball.

Let be a real-valued function defined on an interval and let . The four one-sided limits(1)(2)(3)and(4)are called the Dini derivatives of at . Individually, they are referred to as the upper right, lower right, upper left, and lower left Dini derivatives of at , respectively, and any or all of the values may be infinite.It turns out that continuity at a point of a single Dini derivative of a continuous function implies continuity of the other three Dini derivatives of at , equality of the four Dini derivatives, and (usual) differentiability of the function . In addition, the Denjoy-Saks-Young theorem completely characterizes all possible Dini derivatives of finite real-valued functions defined on intervals and--as corollaries--the Dini derivatives of all monotone and continuous functions defined on intervals.Many other important properties of Dini derivatives have been studied and characterized. Banach showed that the Dini derivative..

Let be a contraction mapping from a closed subset of a Banach space into . Then there exists a unique such that .

Let and be Banach spaces and let be a function between them. is said to be Gâteaux differentiable if there exists an operator such that, for all ,(1)The operator is called the Gâteaux derivative of at . is sometimes assumed to be bounded, though much of the theory of Gâteaux differentiability remains unchanged without this assumption.If the Gâteaux derivative exists, it is unique.A basic result about Gâteaux derivatives is that is Gâteaux differentiable at a point if and only if all the directional operators(2)exist and form a bounded linear operator . In addition, the Gâteaux derivative satisfies analogues of many properties from basic calculus including a mean-value property of the form(3)One definition of the Fréchet derivative pertains to uniform existence of the Gâteaux derivative on the unit sphere of (Andrews and Hopper). In particular, then, Fréchet differentiability..

The word argument is used in several differing contexts in mathematics. The most common usage refers to the argument of a function, but is also commonly used to refer to the complex argument or elliptic argument.An argument of a function is one of the parameters on which the function's value depends. For example, the sine is a one-argument function, the binomial coefficient is a two-argument function, and the hypergeometric function is a four-argument function.

The term analysis is used in two ways in mathematics. It describes both the discipline of which calculus is a part and one form of abstract logic theory.Analysis is the systematic study of real and complex-valued continuous functions. Important subfields of analysis include calculus, differential equations, and functional analysis. The term is generally reserved for advanced topics which are not encountered in an introductory calculus sequence, although many ideas from those courses, such as derivatives, integrals, and series are studied in more detail. Real analysis and complex analysis are two broad subdivisions of analysis which deal with real-values and complex-valued functions, respectively.Derbyshire (2004, p. 16) describes analysis as "the study of limits."Logicians often call second-order arithmetic "analysis." Unfortunately, this term conflicts with the more usual definition of analysis..

An algebraic function is a function which satisfies , where is a polynomial in and with integer coefficients. Functions that can be constructed using only a finite number of elementary operations together with the inverses of functions capable of being so constructed are examples of algebraic functions. Nonalgebraic functions are called transcendental functions.

Let be a vector space over a field , and let be a nonempty set. Now define addition for any vector and element subject to the conditions: 1. . 2. . 3. For any , there exists a unique vector such that . Here, , . Note that (1) is implied by (2) and (3). Then is an affine space and is called the coefficient field.In an affine space, it is possible to fix a point and coordinate axis such that every point in the space can be represented as an -tuple of its coordinates. Every ordered pair of points and in an affine space is then associated with a vector .

Dini's theorem is a result in real analysis relating pointwise convergence of sequences of functions to uniform convergence on a closed interval.For an increasing sequence of continuous functions on an interval which converges pointwise on to a continuous function on , Dini's theorem states that converges to uniformly on .

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