On a property of plane curves (Q1034046)

For a continuous curve \(\gamma: [0,1]\to [0,1]^2\) such that \(\gamma(0)= (0,0)\) and \(\gamma(1)= (1,1)\), it is proved that, for each natural number \(n\) there exists a sequence of points \(A_i\) \((0\leq i\leq n)\) on \(\gamma\) such that \(A_0= (0,0)\), \(A_{n+1}= (1,1)\), and the sequences \(\overarrow{\pi({A_iA_{i+1}})}\) and \(\overarrow{\pi_2({A_iA_{i+1}})}\) \((0\leq i\leq n)\) are positive and have the same up order, where \(\pi_1\) and \(\pi_2\) are projections on the \(x\)-axis and \(y\)-axis, respectively.