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Meromorphic functions, i.e. quotients of holomorphic functions, are a standard concept of (complex) analysis.

What can be said about quotients of real analytic functions? Do they constitute a function field, too? It seems to be a natural idea, but I have not even found a name for that construction.

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Quotient of two real-analytic functions is real-analytic whenever denominator is not zero. You can find the proof here: https://www.math.ucdavis.edu/~hunter/intro_analysis_pdf/ch10.pdf (pages 10-11)

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