Let be a real-valued differentiable function on an open subset of . Assume that and that for all and is also in . Suppose that is homogeneous of degree , in the sense that for all and . By means of the Chain Rule or otherwise, show that
for all . (Here denotes the derivative of at , viewed as a linear map .)
Conversely, show that any differentiable function on with for all must be homogeneous of degree .