Do holes affect the type of function (even, odd or neither)

Question

We can know whether a function is even or odd by substituting using F(-x) but what if the function has a single hole like this

f(x) = $\frac{x(x-2)}{x-2}$

is such a function considered odd or neither? since substitution in the original function will confirm that it is neither while substitution after simplification will confirm that it is odd

Answer 1

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.