Image of an accumulation point under a continuous function Let $X$ be a compact Hausdorff topological space and let $f:X \to F$ ($F$ can be assumed metric space) be a continuous map. Let $\left(x_n\right)$ be a sequence in $X$. Now, there are two possibilities:

*

*$\left(x_n\right)$ is eventually constant, i.e., $x_n=x_0$ for all but finitely many $n$.


*We can think of $\left(x_n\right)$ as an infinite set in $X$, which has an accumulation point in $X$, say $x_0$.
Now, what can we say about $f(x_0)$?
Can we say that the sequence $(f(x_n))$ converges to $f(x_0)$?
A detailed answer will be of great help. Thanks in advance.

A bijective continuous map from a compact space to a Hausdorff space is a homeomorphism.
Thus every open set $U\subset X$, $f(U)$ is open in F. So, if every open set containing $x_0$ interests X (limit point), the image of those open sets (containing $f(x_0)$ )will also intersect $f(X)$.
