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1. ### Introduction/What should I do?

Talking to the physics advisor is not a bad idea, but I would definitely recommend stopping by room 236 to talk to the students, if you haven't already.
2. ### Prove there's no differentiable f such that f(f(x)) = g(x) with g'(x) < 0 for every x

Anyway, my approach was to apply the mean value theorem to show that f is injective. This implies something contradictory about f'.
3. ### Prove there's no differentiable f such that f(f(x)) = g(x) with g'(x) < 0 for every x

The fixed point theorem requires f to be bounded. Otherwise, take f(x) = x + 1 See http://en.wikipedia.org/wiki/Smooth_function" [Broken] for a differentiable but not C^1 function.
4. ### Prove there's no differentiable f such that f(f(x)) = g(x) with g'(x) < 0 for every x

How do you know that f' is continuous?
5. ### Prove there's no differentiable f such that f(f(x)) = g(x) with g'(x) < 0 for every x

I don't see it. How do you show that f is bounded?
6. ### Prove there's no differentiable f such that f(f(x)) = g(x) with g'(x) < 0 for every x

You want to prove that f is monotone. Then, you want to prove that f isn't monotone.
7. ### Analytic on open strip

Define g_y : \mathbb{R} \rightarrow \mathbb{C} so that g_y(x) = f(x+iy). Cauchy's theorem plus continuity of f at the boundary imply that \int_{-a}^a (g_y(x)+g_y(-x))dx = 0 (taking a symmetric rectangular contour with base arbitrarily close to the real line). The continuity of g_y gives that...
8. ### Complex analysis

Neither. Both poles and essential singularities require the relevant function to be holomorphic on a deleted neighborhood of the singularity. z^{-\frac{1}{2}} isn't even continuous on one of these neighborhoods. In descriptive terms, however, it would look like half of a simple pole stretched...