/sci/ - Science & Math

[Anonymous No.8529971]

Is there any reason to not use the sequential definition of continuity over the epsilon-delta one?

[Anonymous No.8529974]

Yes. But first of all they are equivalent, so using one over the other doesnt really make sense. You can always use one, then [insert proof here] to "use" the other.
The sequential definition has a more complex geometric analog.
The sequential definition is exhaustive, so using it to prove continuity or limits is much more difficult. One would need to assume one sequence does not converge and reach a contradiction, whereas in the latter one can prove it just by asserting inequalities.

[Anonymous No.8529979]

>>8529971
I dunno, whats the sequence definition again?

epsi-delta can be expressed in terms of epsilon balls (=open discs), which are very intuitive, and espi-delta is convenient to work with.

what are the upsides of using sequences?

[Anonymous No.8529987]

>>8529979
epsilon open balls in this context are segments not disks

[Anonymous No.8530004]

>>8529987
they are discs in R^2. and with epsilon balls you may as well define what continuity means in an arbitrary metric space, and there I like to imagine my open balls as discs, but I guess you're right that in the standard scenario of funtions R->R these would just be open intervals