# A question about limit superior for function

It is well known that limit of function can be converted to limit of sequence. I wonder if it still holds for limit superior of function. This problem is formulated as follows: For function $$f:\mathbb R\rightarrow\mathbb R$$ and $$a\in\mathbb R$$, define $${\lim\sup}\limits_{x\to a}f(x)$$ to be $$\inf\limits_{\delta>0}(\sup\limits_{0<|x-a|<\delta}f(x))$$. Can we have $${\lim\sup}\limits_{x\to a}f(x)=c$$ iff $${\lim\sup}\limits_{n\to\infty}f(x_n)=c$$ for any sequence $$<x_n>$$ satisfying 1)$$x_n\in\mathbb R$$, 2)$$x_n\to a$$ and 3)$$x_n\ne a$$. I have no idea how to prove it, can you help me? Thanks!

## Answers and Replies

mathman
It is not true as you stated it. The limsup (x->a) ≤ c for any sequence and = c for at least one sequence.

It is not true as you stated it. The limsup (x->a) ≤ c for any sequence and = c for at least one sequence.
I cannot understand your reply, could you please explain in more detail? Thanks.

mathman
My original intention is to try to establish the statement "both lim sup f(x) and lim inf f(x) exist and equal c (possibly $$\pm\infty$$) iff lim f(x) exists and equals c" from analogical statement for sequence. But now this approach is not feasible. I then proved the above statement for functions by definition. Thank you again, mathman!