But I can definitely assure you that a proper definition of functions means that domain and range is specified.
Secondly,
Maybe before college, you are taught under the assumption that real number is everything. Therefore domain is everything that lies in real number where the function is defined
"Missing Point Discontinuity The discontinuity is said to be of missing point type if the limit of the function exists at point 'a' but the function is not defined at that point, i.e. lim x→a f(x) exists but f(a) is not defined."
"you are taught under the assumption that real number is everything. Therefore domain is everything that lies in real number where the function is defined"
Use this definition to construct a domain for xsin(2/x)
"Missing Point Discontinuity The discontinuity is said to be of missing point type if the limit of the function exists at point 'a' but the function is not defined at that point, i.e. lim x→a f(x) exists but f(a) is not defined."
For now, this definition is good enough for you. A point $a \in \mathbb{R}$ is a limit point of a subset $D \subset R$ iff every open interval in $\mathbb{R}$ containing the point $a$ contains a point of $D$.
Because if this is continuous, for the same reason. $f(x)=x\sin(2/x)$ was continuous on its domain $\mathbb{R} - 0$. This does need a little bit more proof, I haven't verified this exactly properly. But I think it's true most probably.
The discontinuity is said to be of missing point type if the limit of the function exists at point 'a' but the function is not defined at that point, i.e. lim x→a f(x) exists but f(a) is not defined."
It is defined for points specifically outside the domain.