If $x$ is a fixed point of a continuous function $f$, there is an open neighborhood $N$ of $x$ with $f(N)subseteq N$

Let $Omega$ be an open subset of a topological space $E$ and $xinOmega$ be a fixed point of a continuous function $f:Omegato E$.

How can we show that there is an open neighborhood $N$ of $x$ with $f(N)subseteq N$.

Clearly, by continuity, if $N_2$ is an open neighborhood of $f(x)=x$, there is an open neighborhood $N_1$ of $x$ with $f(N_1)subseteq N_2$, but it’s not immediately clear why we can take $N_1=N_2$.