Let $x_0$ be an accumulation point of the set $D \subset \mathbb{R}$. We say that $y$ is a limit point of a function $f:D \rightarrow \mathbb{R}$ in $x_0$ iff there exists a sequence $(x_n)$, where $x_n \in D\setminus \{x_0\}$ for $n\in \mathbb{N}$ and $x_n \rightarrow x_0$, such that $y=\lim_{n\rightarrow \infty} f(x_n)$.
I don't know how to prove the following lemma:
If $f: (0,c) \rightarrow \mathbb{R}$ be continuous then the set of all limits points of $f$ in $0$ is the interval $$[\liminf_{x\rightarrow 0}f(x), \limsup_{x\rightarrow 0}f(x)].$$