Recent content by Sqens

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.

Anyway, my approach was to apply the mean value theorem to show that f is injective. This implies something contradictory about f'.

The fixed point theorem requires f to be bounded. Otherwise, take f(x) = x + 1 See http://en.wikipedia.org/wiki/Smooth_function" for a differentiable but not C^1 function.

How do you know that f' is continuous?

I don't see it. How do you show that f is bounded?

You want to prove that f is monotone. Then, you want to prove that f isn't monotone.

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...

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...