TITLE: Is there a bijective, monotonically increasing, strictly concave function from the reals, to the reals?
QUESTION [6 upvotes]: I can't come up with a single one.
The range should be the whole of the reals. The best I have is $\log(x)$ but that's only on the positive real line. And there's $f(x) = x$, but this is not strictly concave. And $-e^{-x}$ only maps to half of the real line.
Any ideas?

REPLY [3 votes]: $f(x) = \pi x+ \int_0^x \arctan (-t)\,dt$ is an example. Many more examples like this one can be constructed.