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I had an exam this afternoon and a question was:

Is there a function $g:\mathbb R\to\mathbb R$ such that for any function $f:\mathbb R\to\mathbb R$ (not necessarily continous) the function $h=g\circ f$ is continuous ?

I don't think so, but I'm not able to prove it.

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What do you think about the function $g:\mathbb R\to\mathbb R$ defined by $g(x)=0$ for all $x\in\mathbb R$ ?

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What functions $g$ make $g\circ f$ continuous if $f$ is defined as $$f(x)=\begin{cases}\alpha &&\text{if }x=0\\ \beta &&\text{if }x\neq 0\end{cases}$$ for real $\alpha$ and $\beta$? That is, when is $$(g\circ f)(x)=\begin{cases}g(\alpha)&&\text{if }x=0\\g(\beta)&&\text{if }x\neq 0 \end{cases}$$ continuous? When is it continuous for all $\alpha$ and $\beta$?

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