TITLE: Prove or disprove that if $f$ is continuous function and $A$ is closed, then $\,f[A]$ is closed.
QUESTION [6 upvotes]: If $f:X\to Y$ is a continuous function and $A\subset X$ is a closed set, 
then $f[A]$ is also closed.
I know that if $f[A]$ is closed implies $A$ is closed then $f$ is continuous, but I'm not sure that the converse is true, I can't find a counterexample either.

REPLY [7 votes]: Another, and perhaps simpler, counterexample is
$$
f(x)=\frac{x}{1+x}:\overbrace{[0,\infty)}^{\text{closed}}\mapsto\overbrace{[0,1)}^{\text{not closed}}
$$
You'll notice that all the counterexamples consist of unbounded domains. A closed and bounded set in $\mathbb{R}^n$ is compact, and it is true that if $f$ is continuous and $A$ is compact, then $f(A)$ is compact, which is also closed.