Oddness makes the restriction linear not just positively homogeneous

Let g be a real-valued continuous function on the set {$(x,y)\in {\mathbb{R}}^2$ | $x^2+y^2=1$} such that g(0, 1) = g(1, 0) = 0 and g(−x, −y) = −g(x, y). Define $f:{\mathbb{R}}^2\to \mathbb{R}$ by $f(x,y)=\sqrt{x^2+y^2}\cdot g(x/\sqrt{x^2+y^2},y/\sqrt{x^2+y^2})$ if (x, y) ≠ (0, 0), and f(x, y) = 0 if (x, y) = (0, 0). For each $(a,b)\in {\mathbb{R}}^2$, define $h_{(}a,b):\mathbb{R}\to \mathbb{R}$ by h_(a,b)(t) = f(ta, tb). Which of the following statements are necessarily true?