First of all we know, $f(x)=sin(x)$ is an odd function because $f(x)=-f(-x)$
The question is if there is a discontinuity (single , interval ,..etc).
I will give you some examples to get what I mean:
ex1: $f(x)=sin(x)\times\frac{x^2-4}{x^2-4} :x\neq\pm2$
ex2: $f(x)=sin(x)\times\frac{x-1}{x-1} :x\neq1$
ex3: $f(x)=sin(x)\times\frac{(x-1)(x+2)}{(x-1)(x+2)} :x\neq1 ,x\neq-2$
ex4: $f(x)=sin(x)\times\frac{\sqrt{x^2-1}}{\sqrt{x^2-1}} :x\gt1$
I feel abit confused because I cant say $f(x)=-f(-x) $ for all $x$, but intuitively it feels ok. is that definition not accurate? is there is something I am missing up?
A function $f$ is odd if whenever both $x$ and $-x$ are in its domain $$ f(x) = -f(-x). $$
We don't usually include the italicized part of the previous definition since it's not usually an issue.
The formal definition of a function requires specifying its domain. Think about the domains of the functions you are trying to define with your formulas.