Proof for Continuous function

first, we have a function: with variables of , (m=1,2,...,M)

we know that the , can be considered as constants.

Now, based on we have a function of that to the maximize the minimum of , by optimally allocating the as

,

where ,

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

where the now becomes a variable which is the sum of the (m=1,2,...,M).

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

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