We say that $f$ is Borel measurable if for $c \in \mathbb{R}$, $\{ x \in E : f(x) > c \}$ is a Borel set.
Let $B$ be a Borel measurable set, and let $f$ be a continuous strictly increasing function, then $f^{-1}(B)$ is a Borel set.
My goal is to first show that the given set is a $\sigma$-algebra, then I need to show that it is in fact Borel measurable.
To show it is a $\sigma$-algebra, this comes from the fact that continuous functions preserve the following:
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$f^{-1}(E_n^c) = (f^{-1}(E_n))^c $.
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$f^{-1}(E_j \setminus E_k) = f^{-1}(E_j) \setminus f^{-1}(E_k)$.
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$f^{-1}(\bigcup E_i) = \bigcup f^{-1}(E_i)$.
I think the last step is just to show that $f^{-1}((a, \infty))$ is Borel measurable.
I'm in a bit of need of inspiration as to how to complete this proof.
Best Answer
As t.b. said, for this part you don't need $f$ to be strictly increasing: the statement holds for all continuous functions. See, for example, https://math.stackexchange.com/q/891568/
To handle this question (stated in the title), consider the inverse function $g=f^{-1}$. It is a continuous function, defined on the range of $f$ (an interval of $\mathbb R$). Apply the first part to $g$.