The derivative, the second derivative, or even the function itself could easily be discontinuous at this point.
But needn't be! See for example f(x) = exp(-1/x) (x > 0), 0 (x ≤ 0).
Wikipedia has an analysis.
(Of course, the space of objects isn't exactly isomorphic to the real line, but it's still a neat example.)
Agreed, but it is not obvious to me that my utility function needs to be differentiable at that point.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.