Remark

Tag 08H2, permanent. Cite it as [Bourbaki, Tag 08H2].

Functions of a Real Variable, chapter V, Appendix 0. Hardy Fields. (H) Functions

The conditions imposed on $q(x)$ may not be improved, as the following examples show:

p(x) = e^x, \quad q(x) = 1 = \frac{p(x)}{p'(x)}, \quad p(x+q(x)) = ep(x)

p(x) = \log x, \quad q(x) = x - \log x \ll \frac{p(x)}{p'(x)} = x \log x,

p(x-q(x)) = \log \log x \ll p(x).

We shall first study the inverses of (H) functions of a particular sort:

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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