Comparing the ranges of continuous functions (Q1189108)

The paper presents the two following theorems concerning functions fulfilling the \(N\) condition of Lusin: If \(f\) and \(g\) are differentiable almost everywhere (a.e.) on the unit interval \(I\), \(| f'(x)|\geq| g'(x)|\) a.e. on \(I\) and the equation \(f(u)=f(v)\) implies the equation \(g(u)=g(v)\), then \(m(f(I))\geq m(g(I))\), where \(m\) denotes Lebesgue outer measure. If \(| f'(x)|=| g'(x)|\) a.e. on \(I\) and if each of the equations \(f(u)=f(v)\) and \(g(u)=g(v)\) implies the other, then either \(f+g\) or \(f-g\) is constant on \(I\).