I'm working independently through Spivak's Calculus on Manifolds and I've come across a stumbling block with respect to two of his questions.
The first question is
2.22. If $f:\mathbb R^2\longrightarrow\mathbb R$ and $D_2f=0$, show that $f$ is independent of the second variable.
$D_2f$ is the second partial derivative of $f$, defined by $$D_2f =\lim_{h\rightarrow0}\frac{f(a^1,a^2+h)-f(a^1,a^2)}{h}$$ where $a\in\mathbb R^2$ with $a=(a^1,a^2)$.
A function $f$ is independent of the second variable if $\forall y_1,y_2\in\mathbb R, f(x,y_1)=f(x,y_2)$
Then, we have a second question
2.23(b). Let $A = \{(x,y):x<0, $ or $ x\ge0$ and $y\ne 0\}$. Find a function $f: A\longrightarrow\mathbb R$ such that $D_2f = 0$, but $f$ is not independent of the second variable.
Clearly, $A\subset\mathbb R^2$. Any such function we define over $A$, we can extend to all of $\mathbb R^2$, fulfill the conditions of 2.22, and arrive at a contradiction. What am I missing here?
What's really going on can be explained in terms of derivatives of one-variable functions. If $I$ is an interval, $f:I\to\mathbb R$ and $f'=0$ then $f$ is constant. But that depends on $I$ being an interval. Say $E=\mathbb R\setminus\{0\}$. Define $f:E\to\mathbb R$ by $f(x)=1$ for $x>0$, $f(x)=-1$ for $x<0$. Then $f'=0$ everywhere on $E$ but $f$ is not constant.
For the actual example asked for in the exercise: Say $f(x,y)=0$ for $x<0$. For $x\ge 0$, let $f(x,y)=x^2$ for $y>0$, $-x^2$ for $y<0$. Then $f$ is differentiable in $A$, $D_2f=0$ but $f$ is not independent of $y$.