# 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!

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.

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!