Theorem. If the function $f$ of the real variables $x_1,\,\ldots,\,x_k$ satisfies the identity

(1)

Proof. Let $f(tx_1,\,\ldots,\,tx_k) := \varphi(t)$ . Differentiating with respect to $t$ we obtain $$\varphi'(t) = x_1f'_{x_1}(tx_1,\,\ldots,\,tx_k)+\ldots+x_kf'_{x_k}(tx_1,\,\ldots,\,tx_k) = \frac{1}{t}[tx_1f'_{x_1}(tx_1,\,\ldots,\,tx_k)+\ldots+tx_kf'_{x_k}(tx_1,\,\ldots,\,tx_k)],$$ which by (1) may be written $$\varphi'(t) = \frac{n}{t}f(tx_1,\,\ldots,\,tx_k) = \frac{n}{t}\varphi(t).$$ Accordingly, $$\frac{\varphi'(t)}{\varphi(t)} = \frac{n}{t},$$ which implies the integrated form $$\ln|\varphi(t)| = \ln{t^n}+\ln{C}$$ for any positive $t$ . Thus we have $\varphi(t) = Ct^n$ , where $C$ is independent on $t$ . Choosing $t = 1$ we see that $C = \varphi(1)$ , and therefore $\varphi(t) = t^n\varphi(1)$ . This last equation means that $$f(tx_1,\,\ldots,\,tx_k) = t^nf(x_1,\,\ldots,\,x_k)$$ saying that $f$ is a (positively) homogeneous function of degree $n$ .

Bibliography

1

ERNST LINDELÖF: Differentiali- ja integralilasku ja sen sovellutukset II. Mercatorin Kirjapaino Osakeyhtiö, Helsinki (1932).