I'm studying the following proof by I'm not able to understand the main step. Let \begin{equation} \sigma(x)=\sqrt{\beta x(N-x)+\alpha x} \end{equation} defined for $x \in [0, N+\frac{\alpha}{\beta}]$. I want to prove that there exist a constant $L > 0$ such that \begin{equation} |\sigma(x)-\sigma(y)| \le L \sqrt{|x-y|} \quad \forall x \in [0, N+\frac{\alpha}{\beta}] \end{equation} The proof the I'm studying works as follow:
choose $\varepsilon=\frac{1}{4}(N+\frac{\alpha}{\beta})$, and consider separately the regions \begin{align}\varepsilon &\le x \le N+\frac{\alpha}{\beta}- \varepsilon, \quad \varepsilon \le y \le N+\frac{\alpha}{\beta}- \varepsilon \\ 0 & \le x,y \le \varepsilon\\ N& +\frac{\alpha}{\beta}- \varepsilon \le x,y \le N+\frac{\alpha}{\beta},\\ N& +\frac{\alpha}{\beta}- \varepsilon <x \le N+\frac{\alpha}{\beta},\quad 0<y<\varepsilon \\ 0&<x<\varepsilon,\quad N +\frac{\alpha}{\beta}- \varepsilon <y < N+\frac{\alpha}{\beta} \end{align} Then it is straightforward to show that the previous relation holds.
But why holds? what is the problem in the original domain?