Negating Limits and Continuity

[Treadstone 71]

Let f:I->R and let c in I. I want to negate the statements: "f has limit L at c" and "f is continuous at c". Are these correct?

f does not have limit L at c if there exists e>0 such that for some sequence {x_n} converging to c, |f(x_n)-L|>e for every n.

f is not continuous at c if there exists e>0 such that for some sequence {x_n} converging to c, |f(x_n)-f(c)|>e for every n.

edit: also, what is the negation of "f has a limit at c"?

 

[HallsofIvy]

Yes, those will work, since "for all sequences {a_n} converging to c, the sequence {f(x_n)} converges to L", though not the standard definition of limit, is equivalent to it.

As far as "f does not have a limit at c" (as opposed to "f does not have limit L at c"), you just have to add "for every L":
"For every number L, there exist an epsilon> 0 and a sequence {x_n} that converges to c, such that |f(x_n)- L|> epsilon."

 

[Treadstone 71]

Thanks a lot.