My textbook argues that the Cobb-Douglass utility function $u=(x1)^a(x2)^b$ with $a,b>0$ and $a+b<1$ is concave on $R2+$ by computing the Hessian and showing it to be negative semidefinite for all points in $R2+$.
However, I feel this method is flawed because $R2+$ is not an open set. A function is concave on the set $A$ if and only if its Hessian is negative semidefinite for all $x$ in $A$, but the assumption is that $A$ is an open and convex set. This does not hold, so the above methodology seems flawed. I am getting confused about this, so I would really appreciate some help please!
For reference, the textbook I am using is this: https://mjo.osborne.economics.utoronto.ca/index.php/tutorial/index/1/cvn/t