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Is there any notion of taking a continuous product of the values of a real function over an interval? Only thing I could think of was exponentiating to the power of the integral over that interval.

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If in some way we have $\sum a_n\to\int a_n ~ dx$ for a suitable definition of "$\to$"

Then $\prod a_n = \exp\left(\log \prod a_n \right)=\exp\left(\sum \log(a_n)\right)\to\exp\left(\int \log(a_n)~dx\right)$

Edit: after googling I found this

https://en.wikipedia.org/wiki/Product_integral

which might interest you

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