Let $f:X\to X$ be a continuous map and $x\in X$. A point $y\in X$ is called an $\alpha$-limit point of $x$ under $f$ if and only if there is a strictly increasing sequence of positive integer $\{k_n\}_{n=0}^\infty$ and a sequence of points $\{y_n\}_{n=0}^\infty$ such that
1) $f^{k_n}(y_n)= x$
2) $\lim_{n\to \infty} y_n= y$.
Also, a point $y\in X$ is called a non-wandering point, if for every open set $U$ of $y$, there is $n\in\mathbb{N}$ such that $f^n(U)\cap U\neq \emptyset$.
I know that if $f$ is a homeomorphism, then every $\alpha$- limit point of $x$ is a non-wandering point. But I do not know if $f$ is a continuous map, then every $\alpha$- limit point of $x$ is a non-wandering point?
Please help me to know it.