- #1
Danijel
- 43
- 1
When we define a limit of a function at point c, we talk about an open interval. The question is, can it occur that function has a limit on a certain interval, but it's extension does not? To me it seems obvious that an extension will have the same limit at c, since there is already infinitely many points around its limit L, so adding more, or none, doesn't change anything (since we are now looking at sequences whose limit is c, but end up in the extension's domain,which is larger than the original domain, so we have more sequence values, of course, if we don't talk about Cauchy definition). However, my short experience with calculus has made me careful of concluding wacky statements (at least for those Iack I a firm argument).
Edit: I note that a function can behave differently on a different interval, hence it can happen that, for some small number δ, there is such cn for which f(cn) exits the interval <L-δ,L+δ>, and the function (the extension) does not have a limit L at c.
But, since for large n, by our assumption that the sequence cn has a limit at c, for a large n, the f(cn) cannot exit the observed interval, since preimage is already in a small interval around c, and for every such point, it holds that |f(cn)-L|<δ. So a function may have a limit. I really can't think straight.
Edit: I note that a function can behave differently on a different interval, hence it can happen that, for some small number δ, there is such cn for which f(cn) exits the interval <L-δ,L+δ>, and the function (the extension) does not have a limit L at c.
But, since for large n, by our assumption that the sequence cn has a limit at c, for a large n, the f(cn) cannot exit the observed interval, since preimage is already in a small interval around c, and for every such point, it holds that |f(cn)-L|<δ. So a function may have a limit. I really can't think straight.
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