A topological property of the solution funnel of \(x'=f(t,x)\), \(x(0)=x_0\) (Q1612546)

The author considers the solution funnel \(S\) to initial value problems \(x'= f(t,x)\), \(x(0)= x_0\), with \(f:[0,1]\times \mathbb{R}^n\to \mathbb{R}^n\) continuous and bounded. It is known that \(S\) is connected but, in general, not pathwise connected. The author proves that \(S\) is pathwise connected in case that \(f\) is in addition quasi-monotone increasing in \(x\) with respect to the coordinate cone. In particular, if \(n= 1\) it follows that \(S\) is always pathwise connected.