The definition of function convergence is similar to that of sequence convergence. Cauchy Convergence Criterion: On the Convergence Definition of Function f(x) at Point x0. For any real number b>0, c > exists; 0, for any x 1, x2 satisfies 0.

The definition of convergence well embodies the spiritual essence of mathematical analysis.

Given a function sequence defined on the interval I, u 1(x), u2(x), u3(x) ... to the United Nations (x) ... and then the expression U 1 (x)+U2 (x)+U3 (x)+

Extended data

For each determined value X0∈I, the function term series (1) becomes a constant term series U1(x0)+U2 (x0)+U3 (x0)+...+UN (x0)+... (2) This series can converge or diverge. If the series (2) diverges, the point x0 is called the divergence point of the series of function terms (1).

The whole convergence point of the function term series (1) is called its convergence domain, and the whole divergence point is called its divergence domain, corresponding to any number x in the convergence domain. Function term series is called convergent constant term series, so there is a definite sum S.

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