What is the general reason for functions which can only be defined implicitly?
Is this because they are multivalued (in which case they aren't strictly functions at all)?
Is there a proof?
Clarification of question.
The second part has been answered with example of single value function which cannot be given explicitly. The third part was automatically answered because there can't be a proof that necessarily implicit functions are all multivalued by way of the example of one that wasn't.
I don't think that the first part has yet been addressed. Stating that the answer can't be expressed in "elementary functions" seems tantamount to saying that it a necessarily implicit function, unless I'm missing something about the definition of "elementary functions". Such answers seem to imply that the equation could be solved in terms of "non-elementary" functions.If this is correct than I need to find out about them and how they could be used to calculate the dependent variable solely in terms of the independent one (my notion of an explicit function). This would seem to violate the notion of a function which could only be defined implicitly. I am also not concerned with whether or not the solution is closed or open form (by which I mean finite number of terms or infinite).