Given a smooth Riemannian manifold (M,g), is the function $f:M \rightarrow \mathbb{R}$, defined as the maximum of sectional curvatures at each point $m \in M$, smooth?
I think for each point $m \in M$, sectional curvature is a real valued function on the Grassmannian(2,d) where $d$ is the dimension of $M$. Since the Grassmannian is compact, maximum is achieved and $f$ can be defined.
I would like to learn more about the function $f$; thank you.