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Let u : R × M → R {\displaystyle u:\mathbb {R} \times M\to \mathbb {R} } be any continuous function. Then the timewise-max function of u {\displaystyle u} is a map f : R → R {\displaystyle f:\mathbb {R} \to \mathbb {R} } defined by:
f ( t ) = sup x ∈ M u ( t , x ) {\displaystyle f(t)=\sup _{x\in M}u(t,x)}
When M {\displaystyle M} is a compact manifold, we can replace sup {\displaystyle \sup } by max {\displaystyle \max } .
