 Solution

Assume the homogeneous function is differentiable. Applying the multivariable chain rule to $f(tx,ty)=t^{m}f(x,y)$ gives, for $t$ near $1$,Setting $t=1$ proves Euler theorem for homogeneous functions:Only positive $t$ near $1$ are needed, so the derivation remains valid even when a noninteger $m$ makes the printed phrase “all real $t$” require qualification.