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) seperable, 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 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.

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.

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, additively (or multiplicatively) seperable, 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 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.

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) seperable, 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 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.