Can a function be considered odd or even if has a discontinuity?
Solution 1:
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.