Proof for Continuous function

first, we have a function: $\displaystyle y_k $ with variables of $\displaystyle x_m $, (m=1,2,...,M)

$\displaystyle y_k = \frac{\sum_{m=1}^M x_m \alpha_{m,k}}{\sum_{m=1}^M x_m \beta_{m,k} +1}$

we know that the $\displaystyle x_m \geq 0 $, $\displaystyle \alpha_{m,k}, \beta_{m,k}$ can be considered as constants.

Now, based on $\displaystyle y_k $ we have a function of $\displaystyle z$ that to the maximize the minimum of $\displaystyle y_k $,$\displaystyle (k=1,2,...,K)$ by optimally allocating the$\displaystyle x_m $ as

$\displaystyle z = \max_{x_m} \min_{k=1.\cdot,K} (y_k).$,

where $\displaystyle \sum_{m=1}^M x_m = x$,

Actually, we can alternatively express the above function of z as

$\displaystyle z=f(x),$

where the $\displaystyle x$ now becomes a variable which is the sum of the $\displaystyle x_m$ (m=1,2,...,M).

then, we want to prove $\displaystyle z=f(x),$ is continuous to x and the function f(') returns the maiximized minimum value of $\displaystyle y_k $ with an input variable of $\displaystyle x$ .

for the z=f(x), I have prove it is a strictly monotonically increasing function to x, but I failed to prove the continuity of it.

Thanks