Do holes affect the type of function (even, odd or neither)
It depends upon how you defined odd function and even function. Let $D\subset\Bbb R$.
- If you say that a function $f\colon D\longrightarrow\Bbb R$ is odd if $x\in D\implies-x\in D$ and $f(-x)=-f(x)$ (this would be my definition), then $f$ is not odd (since $-2$ belongs to its domain, but $2$ doesn't).
- If you say that a function $f\colon D\longrightarrow\Bbb R$ is odd if, whenever both $x$ and $-x$ belong to $D$, then $f(-x)=-f(x)$, then your function is odd.