Convergence must be bounded, but not necessarily; Continuity means that the function is an uninterrupted curve within a certain range. It is not necessarily related to convergence and boundedness.

For example, a sequence is a typical discontinuous function, but it can converge and be bounded; Y=sinx is a typical bounded and convergent continuous function.

Let {an} be a sequence and a be a fixed real number, if for any given b >；; 0, with a positive integer n, so that for any n >；; N, you have |-a | < B is a constant, that is, the sequence {an} converges to a (the limit is a), that is, the sequence {an} is a convergent sequence.

Extended data

For any X0∈[a, b], the sequence of points generated by the iterative formula Xk+ 1=φ(Xk) converges, that is, when k→∞, the limit of Xk tends to X*, then Xk+ 1=φ(Xk) is called in [a, b].

If X* exists in a neighborhood and r = {x || x-x *|| < δ}}, the point sequence generated by Xk+ 1=φ(Xk) converges for any X0∈R, so Xk+ 1=φ(Xk) converges. ..

Regarding the boundedness of functions, we should pay attention to the following two points:

The (1) function is bounded or unbounded in a certain interval, and both must belong to one;

(2) It is easy to judge whether a function is bounded from a geometric point of view. If two straight lines parallel to the X axis cannot be found, so that the graph of the function is between them, then the function must be unbounded.

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