Finding a function of two variables satisfying some condtions

Question

I am trying to find a function of two variables $f(x, y): [0, \infty) \times \mathbb{R} \to \mathbb{R}$, satisfying the following conditions:

(i) $f(0, y) = 1$ for all $y \in \mathbb{R}$;

(ii) $f(x, y) > 0$ for all $x \in (0, \infty)$ if $y \geq 0$;

(iii) $f(x, y) < 0$ for all $x \in (0, \infty)$ if $y < 0$;

Does there exist a function f(x, y) which satisfies the conditions (i)-(iii)?

Thank you very much for your support.

Answer 1

You can consider the function $f$ defined as $$ f (x, y) = 1 \quad \text{if } x=0 \text{ for all } y \in \mathbb{R}$$ $$ f (x, y) = 1 \quad \text{if } x>0 \text{ and } y \geq 0$$ $$ f (x, y) = -1 \quad \text{if } x>0 \text{ and } y < 0$$

Answer 2

$${1\over2}(2\operatorname{heavi}(x)-\operatorname{kron\delta}(x))(2\operatorname{heavi}(y)+\operatorname{kron\delta}(y)-1)+\operatorname{kron\delta}(x) ,$$where $$\operatorname{heavi}$$is the Heaviside function, &$$\operatorname{kron\delta}$$the Kronecker $\delta$ function.

This could be expressed as a limit of continuous functions: $$\lim_{a\to\infty}{1\over2}(\tanh(ax)-\exp(-(ax)^2)+1)(\tanh(ay)+\exp(-(ay)^2)+\exp(-(ax)^2) .$$or$$\lim_{a\to\infty}{1\over2}(\tanh(ax)-\operatorname{sech}(ax)+1)(\tanh(ay)+\operatorname{sech}(ay))+\operatorname{sech}(ax) .$$