Originally Posted by

**clover616** sorry about my ambiguous definition of $\displaystyle \alpha_{m,k},\beta_{m,k}$.

actually, in the function of $\displaystyle y_k$, the $\displaystyle \alpha_{m,k},\beta_{m,k}$ are predetermined in my model regardness of the $\displaystyle x_m$ or we can say $\displaystyle \alpha_{m,k},\beta_{m,k}$ are independent to the $\displaystyle x_m$.

please have a look of my proof for the monotonicity of the funtion, then may be you can know about the problem.

In my problem , I need to focus on the continuity between $\displaystyle z$ and $\displaystyle x$, instead of that between $\displaystyle z$ and $\displaystyle x_m$ and there is no closed form solution for $\displaystyle x_m$, so this make the problem difficult.