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I am looking for resources on fibres of continuous functions from $\mathbb R$ to $\mathbb R$. Here by fibres of a real valued function $f$, I mean the set $f^{-1}(y)$ for $y\in \mathbb R$. For context, I am trying to determine sufficient conditions on $f$ such that for every cardinal $\lambda$, the set $\{y\in \mathbb R:\,|f^{-1}(y)|=\lambda\}$ is borel.

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  • $\begingroup$ You only need to consider the case $\lambda\le\aleph_0\lor \lambda=\beth_1$. See here. $\endgroup$ Commented Jun 25 at 8:21

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