fn, are continuous on [0,1] and converges pointwisely to f,
must f be continuous ?
There are some 's and 's that should be in there somewhere, but I'm too lazy to find out where, and they shouldn't make any difference anyway. As , the non-integral part clearly goes to 0, and then you might be able to prove by induction that everything is 0. Or something.
The function is symmetric about .
It remains to show that expression goes to zero (such as using Stirling's approximation).
Notice something interesting, there is a power series:
Therefore, if .
But in our case to get .
However, cannot be analytically continued at since explodes that at point.
Somethings seems to be wrong.