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If f is a continuous real-valued function on [a,b] and if any epsilon>0 is given, then there exists a polynomial p on [a,b] such that |f(x)-P(x)|<epsilon for all x in [a,b]. ...

A function, continuous in a finite closed interval, can be approximated with a preassigned accuracy by polynomials. A function of a real variable which is continuous and has ...

The Blancmange function, also called the Takagi fractal curve (Peitgen and Saupe 1988), is a pathological continuous function which is nowhere differentiable. Its name ...

A set in R^n which can be reduced to one of its points, say P, by a continuous deformation, is said to be contractible. The transformation is such that each point of the set ...

Newton's method for finding roots of a complex polynomial f entails iterating the function z-[f(z)/f^'(z)], which can be viewed as applying the Euler backward method with ...

A universal differential equation (UDE) is a nontrivial differential-algebraic equation with the property that its solutions approximate to arbitrary accuracy any continuous ...

A bounded operator T:V->W between two Banach spaces satisfies the inequality ||Tv||<=C||v||, (1) where C is a constant independent of the choice of v in V. The inequality is ...

A function with k continuous derivatives is called a C^k function. In order to specify a C^k function on a domain X, the notation C^k(X) is used. The most common C^k space is ...

If a function f(x) is continuous on a closed interval [a,b], then f(x) has both a maximum and a minimum on [a,b]. If f(x) has an extremum on an open interval (a,b), then the ...

Given two topological spaces M and N, place an equivalence relationship on the continuous maps f:M->N using homotopies, and write f_1∼f_2 if f_1 is homotopic to f_2. Roughly ...