It is not generally possible to restrict $f$ to the class $D$.
Then $F$ restricts to a $C^0$ flow on $M$.
Theorem 2 still holds for $A(x)$ provided that $k$ is restricted to the range $[0,1]$.
The supremum in (1) does not change if we restrict ourselves to rational points in $Q$.
We shall restrict the discussion to plane regions.
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