Consider a function $f:\mathbb{R}_+^2\rightarrow\mathbb{R}$ of two non-negative real variables (or more generally of several real variables) that is increasing in each argument, continuous, additively (or multiplicatively) separable, that is, it can be written in the form $$f(x,y)=a(x)+b(y)$$ for functions of one variable, $a$ and $b$, and satisfies the following notion of "scale invariance": For each $\lambda>0$ and $(x,y),(\tilde{x},\tilde{y})\in\mathbb{R}_+^2$ we have $$f(x,y)=f(\tilde{x},\tilde{y})\Leftrightarrow f(\lambda x,\lambda y)=f(\lambda \tilde{x},\lambda \tilde{y}).$$ Is there an example of such a function that is not everywhere differentiable? We tried hard to find such an example. For example, we know that if such functions exist, they have to be non-differentiable on a dense null set, by the scale invariance property.

Edit: The original question was asked for non-continuous functions and it was answered negatively. In the edited version we ask the question for continuous functions.