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A very simple example of a useful variable change can be seen in the problem of finding the roots of the sixth-degree polynomial: + = Sixth-degree polynomial equations are generally impossible to solve in terms of radicals (see Abel–Ruffini theorem). This particular equation, however, may be written () + = (this is a simple case of a ...
Continuity of real functions is usually defined in terms of limits. A function f with variable x is continuous at the real number c, if the limit of (), as x tends to c, is equal to (). There are several different definitions of the (global) continuity of a function, which depend on the nature of its domain.
The textbook example of a periodic motion is the undamped pendulum. If the phase space is periodic in one or more coordinates, say u ( t ) = u ( t + Ω ) {\displaystyle \mathbf {u} (t)=\mathbf {u} (t+\Omega )} , with Ω {\displaystyle \Omega } a vector [ clarification needed ] , then there is a second kind of periodic motions defined by
Some properties of complex-valued functions (such as continuity) are nothing more than the corresponding properties of vector valued functions of two real variables. Other concepts of complex analysis, such as differentiability , are direct generalizations of the similar concepts for real functions, but may have very different properties.
Continuous function; Absolutely continuous function; Absolute continuity of a measure with respect to another measure; Continuous probability distribution: Sometimes this term is used to mean a probability distribution whose cumulative distribution function (c.d.f.) is (simply) continuous.
In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. [2]
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