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Suppose f, g : R → R are continuous. Determine if the set $E = \{x\;|\; f(x) = g^2(x) + 2020\}$ is open, closed or not necessarily open or closed in R.

I know that the function need not necessarily be open, as I was able to give a counterexample. However, I am not able to come up with a counterexample to show that the set need not be closed (I have a hunch that the set is always closed). However, I am unable to prove that the set would be definitely closed. Please help.

TonyK
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Srinivas Thirumalai
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    $g^2(x)$ can mean either $g(g(x))$ or $(g(x))^2$. Better not to use it. (But I don't think the difference is relevant for this question $-$ $g^2(x)+2020$ is continuous in both interpretations.) – TonyK Oct 31 '20 at 13:55
  • One of the most used definition of continuity for a function is that if $f : A \to B$ is a function between two topological spaces, $f$ is continue if and only if for every open subset $V \subset B$, $f^{-1}(B)\subset A$ is open in $A$. You can show it is equivalent to the fact that for every closed subset $F\subset B$, $f^{-1}(F) \subset A$ is closed. Now, what can you say, if $f,g$ are continuous, about the function $f - g\times g$? And what can you say about the subset ${2020}\subset \mathbb{R}$? – Didier Oct 31 '20 at 13:59
  • Take a look at https://math.stackexchange.com/questions/437829/the-preimage-of-continuous-function-on-a-closed-set-is-closed, along with the fact that difference between two continuous functions is again continuous – Sten Oct 31 '20 at 13:59

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If $h=f-g^2$, then $h$ is continuous too. and $E=h^{-1}\bigl(\{2020\}\bigr)$. So, $E$ is a closed set, since is the pre-image of a closed set with respect to a continuous function.

José Carlos Santos
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