For measurable $f: \mathbb{R} \rightarrow \mathbb{R}$ prove $f(x)$ and $\frac{1}{f(1/x)}$ cannot both be Lebesgue integrable.

First question on MSE! I'd appreciate hints, theorem suggestions, or method suggestions regarding the question in the title or below. Please avoid full solutions. I'm studying for an exam coming up and got stuck on this question:

Problem Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a measurable function. Prove that $f(x)$ and $\frac{1}{f(1/x)}$ cannot both be Lebesgue integrable.

I've taken courses based on and read from Royden & Fitzpatrick if that helps with suggestions.

My attempts so far have focused on trying to find contradictions assuming $f$ is integrable: i.e. $\int_{\mathbb{R}} |f| < \infty$ and defining $S_0 := \{x \in \mathbb{R} | f(x) = 0 \}$. I'm thinking that something is happening with zeros and infinities that destroys the measurability of the alternative function.

Thanks in advance!

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