Real Function/Continuity in a point/Characterization/Fact/Proof

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Suppose that (1) is fulfilled and let be a sequence in converging to . We have to show that

holds. To show this, let be given. Due to (1) there exists a fulfilling the estimation property (from the definition of continuity) and because of the convergence of to there exists a natural number such that for all the estimate

holds. By the choice of we have

so that the image sequence converges to .

Suppose now that (2) is fulfilled. We assume that is not continuous. Then there exists an such that for all there exist elements such that their distance to is at most , but such that the distance of their value to is larger than . This holds in particular for every , . This means that for every natural number there exists a with

This sequence converges to , but the image sequence does not converge to , since the distance of its members to is always at least . This contradicts condition (2).