Showing that a function is constant Let $f:\mathbb{R}^2 \rightarrow \mathbb{R}$ be differentiable at any point. Assume that  $x \cdot f_x(x,y) + y \cdot f_y(x,y) = 0$ for each point $(x,y) \in \mathbb{R}^2$. Show that $f$ is constant in $\mathbb{R}^2$.
Is the following proof correct:
Let $(x_0,y_0) \in \mathbb{R}^2$ and define $g:\mathbb{R} \rightarrow \mathbb{R}^2$ by $g(t)=(tx_0,ty_0)$. $g$ is differentiable at any point so by the chain rule we have $(f \circ g)'(t) = D_{f \circ g} = D_f(g(t)) \cdot D_g(t) = x_0f_x(tx_0,ty_0) + y_0f_y(tx_0,ty_0)$. For $t>0$ we have $(f \circ g)'(t) = \frac1t(tx_0f_x(tx_0,ty_0) + ty_0f_y(tx_0,ty_0)) = 0$. So $f \circ g$ is constant for $t>0$ assume $f \circ g = c$. $f$ is differentiable and thus continuous so we have $c = \lim_{t \to 0^+} f(tx_0,ty_0) = f(0,0)$. 
So if $(x,y) \in \mathbb{R}^2$ we can write $(x,y) = (tx_0,ty_0)$ for $(x_0,y_0) \in \mathbb{R}^2$ and $t>0$ and by what we showed $f(x,y) = f(tx_0,ty_0) = f(0,0)$.
 A: Your proof appears correct to me. If I were writing it, I'd slightly change the phrasing at three points: 
First, I'd say "Let $(x_0, y_0)$ be any point of $R^2$ except the origin, and define ..."
I'd say that because you only use this claim when the point isn't the origin, and as another commenter pointed out, in the case where it is the origin, your further statements seem to say nothing. So avoid confusing the reader. :)
Second, I'd say "For $T \ne 0$" rather than "For $t > 0$". 
Why? Because all you need is non-zero-ness. It's true that soon you'll take a one-sided limit, but when I read the proof, I was puzzled and thought "isn't this true for $t < 0$ as well?" 
Third, I'd say "So $f \circ g$ is constant (although the constant may depend on $(x_0, y_0)$). For $t > 0$, assume that $f \circ g = c_{(x_0, y_0)}$. We'll now show that this is in fact independent of $(x_0, y_0)$."
The two key things here are (1) splitting into two sentences, and (2) explaining the subtlety, so that your reader knows where you're headed.
In fact, I'd start the proof by saying "The broad idea is to show that (i) $f$ is constant along rays through the origin, and (ii) that the constant value is the same for any two rays, namely $f(0,0)$."
