“Theorem on Function Values”

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0.38 Theorem Concerning the Values of a Function_All the values which can be assumed by a function of any number of variables \(f(x,y,z\ldots)\) are given by the formula_\[abc\ldots k+u(a+b+c+\ldots+k),\]_in which \(u\) is absolutely indeterminate, and \(a,b,c\ldots,k\) are the coefficients of the development of \(f\).__Demonstration.–_It is sufficient to prove that in the equality\[f(x,y,z\ldots)=abc\ldots k+u(a+b+c+\ldots+k)\]\(u\) can assume all possible values, that is to say, that this equality, considered as an equation in terms of \(u\), is indeterminate.In the first place, for the sake of greater homogeneity, we may put the second member in the form\[u^{\prime}abc\ldots k+u(a+b+c+\ldots+k),\]for\[abc\ldots k=uabc\ldots k+u^{\prime}abc\ldots k,\]and\[uabc\ldots k


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