A function exists for all values of the independent variable in the domain of the function. A function does not exist for all values of the independent variable excluded from the domain of the function.
For example: does not exist for but does exist for all other real number values of .
is defined, and therefore exists, and is not defined, and therefore does not exist .
On the other hand:
is defined, and therefore exists,
Note: is the set of all real numbers and is the set of complex numbers.
John
My calculator said it, I believe it, that settles it