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).
Best Answer
Not necessarily. Consider the graph $G$ in ${\mathbb R}^2$ of the points $(x,y)$ such that $$ y^5+16y-32x^3+32x=0. $$ This example comes from the nice book "The implicit function theorem" by Krantz and Parks.
Note that this is the graph of a function: Fix $x$, and let $F(y)=y^5+16y-32x^3+32x$. Then $F'(y)=5y^4+16>0$ so $F$ is strictly increasing. Since $F(y)\to-\infty$ as $y\to-\infty$ and $F(y)\to\infty$ as $y\to\infty$, the equation $F(y)=0$ has at least one solution (by the intermediate value theorem), but then it has precisely one. This means that the graph $G$ describes a function (in the strict, traditional sense, not a multivalued function).
However, there is no formula (in the traditional sense) giving us $y$ explicitly as a function of $x$.
This specific example shows one of the reasons why this may happen. Here, we have $y$ as a solution of a quintic equation, but in general there is no algebraic formula that gives us these solutions; this is a deep result of Abel and Galois.
There are similar results stating that certain equations do not have elementary solutions (for example, certain differential equations), so they may define a function but we may not have a "formula" for it. In fact, the concept of what a "function" is underwent a few revisions before reaching its current modern form, precisely as we realized the need for versions more general than "given by an explicit formula". A nice account of the evolution of the term is in the paper by Penelope Maddy, "Does $V$ equal $L$?", J. Symbolic Logic 58 (1993), no. 1, 15–41. jstor, doi: 10.2307/2275321