This is my first post on math help forum, and I feel quiet enthusiastic about it, although the problem I have is pretty challenging.

So let me start. I have a real function (function of a real variable) in form:

wherekis some real constant, andg(x)is an arbitrary unknown also real function. I want to make another functionh(x) in a following way:

which should be smooth (all derivatives exist) at every point. Of course iff(x)is smooth, by definitionh(x) will be also smooth in all points butx=0. To make it also smooth at zero the following condition should be satisfied.

If we consider its Taylor series with this condition,f(x) is either going to be a constant or non-analytic function. Yes, you guessed, I need this second one.

Finally we came to the question.

I should constructg(x) such thath(x) is smooth at every point (zero is the non-trivial one).

I started calculating higher order derivatives off(x). Based on obtained results I reckon that if I findg(x) that satisfies following two conditions, my problem will be solved.

Polynomial functions do not work here, so some trick with exponential one perhaps should work. Anyhow, if somebody has any clue how to proceed with this, euphemism is that I would be very grateful. Of course, I would be happy if someone proves me that this what I am looking for is not possible. I'll simply forward your message to my supervisor , and multiply myf(x) with:

Cheers.