What is the definition of a stadium curve and does it have a curvature that is defined and continuous at each of its points?
I do not know if there exists a general definition of a stadium curve, but a Bunimovich stadium is a well-known example: It is a rectangle capped by semicircles. See http://en.wikipedia.org/wiki/Dynamical_billiards#Bunimovich_stadium
"... the classical stadium [curve] with the boundary that consists of two semicircles and two parallel segments tangent to them ..." from
It's continuous, so is its derivative, but its second derivative is (I believe) not continuous. So I'd say it was in differentiability class C^1.