Suppose $f=f(x_1,x_2,…,x_n)$ is a homogeneous function
$$f(Cx_1,Cx_2,…,Cx_n)=C^\lambda f(x_1,x_2,…,x_n)$$
1) Is $f$ differentiable w/respect to all its arguments?
2) Is $f$ differentiable w/respect to all its arguments at the origin?
3) Is $f$ differentiable w/respect to all its arguments at a point, where some of the $x_i$ are negative or zero?
4) Are all partial derivatives of $f$ also differentiable, to what order and where?
Perhaps Euler's theorem for homogeneous functions is related to the question…
When is a homogeneous function also differentiable?
Best Answer
Not always. Take for example the function
$$f(x,y) = \begin{cases} x & y=0 \\ y & x=0 \\ 0 & \text{else} \\ \end{cases}$$
It is homogenous, but not even continuous, let alone differentiable. The partials in all directions do exist, however, at the origin and everywhere else off-axis. This doesn't make it differentiable at the origin, though.