I am seeking a general mathematical proof & a reference for that proof for something I know intuitively to be true, and can demonstrate by example, but would like to prove. Assume a function with 6 inputs [ f(z,y,x,w,v,u) ] that is non-linear with respect to all six variables. I have five sets of inputs, and the value of at least one variable differs in each set. I calculate the value of f for each of the five input sets, and then calculate the average value of f across all five input sets.

THEN I calculate the average value of u across all five input sets, and substitute the average value of u across the the sets for each of the individual value of u and once again solve for each f and then the average of all f's.

I know that I do not get the same answer averaging the outputs vs using the average input and and then averaging the outputs. I need to know if there is a way to prove that this will always be the case, or to identify the minimum set of assumptions required to assert that it will not be true, or alternatively to identify the set of assumptions that MUST be true in order to get the same result (such that I might infer that if those assumptions are not true, then the results will not be equal).

This is not for school, it is for work. I am at a loss for where to find a proof for something so simple, but I need it to convince a co-worker who does not seem to believe my examples.

thanks