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Since the function defined by (2.23) is continuous and increasing on, it follows that for ; see (2.26).
In order to show that is increasing on it is sufficient to verify that is injective.
Since, from the continuity of, it follows that there is a right neighborhood of the point, in which and hence If there exists a point such that but then is monotonically increasing on It follows from (2.42) that on (2.43).
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com