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 https://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 Loskutov, Alexander; Ryabov, Alexei, Particle dynamics in time-dependent stadium-like billiards, J. Stat. Phys. 108, No. 5–6, 995–1014 (2002). ZBL1124.82310.
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.