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Math and science::Analysis::Tao::09. Continuous functions on R

Function convergence's equivalence to sequence convergence

Let X be a subset of R, let f:XR be a function, let E be a subset of X, let x0 be an adherent point of E, let L be a real, and let ε>0 be a real. Then the following two statements are logically equivalent:

  • f converges to L at x0 in E.
  • For every sequence (an)n=0 which consists entirely of elements of E and converges to x0, the sequence (f(an))n=0 converges to L.

In view of the above proposition, we will sometimes write "f(x)L as xx0 in E" or "f has a limit L at x0 in E" instead of "f converges to L at x0 in E" or limxx0;xEf(an)=L.

Use limits of sequences to calculate limits of functions

This proposition is very important in allowing us to determine what real a function converges to without having to manipulate the definition of function convergence and play around with ε-closeness. Instead we can use all of our built up knowledge of limits of sequences.


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