Let
$ f:X \rightarrow Y$ be a continuous map and $c \in Y$. Is $\{x \in X | f(x)=c\}$ necessarily a closed subset of $X$?
Attempt:
I was thinking about a contradiction. Can this be a counterexample? $f(x) = \frac{1}{x^2+1}$ is continuous on the whole real line and and the image of the set $[0,\infty)$ is the set $(0,1]$. So we have found a closed set, whose image under a continuos function is not closed(and nor open).