Implicit Function Theorem
It can be shown that if R(x, y) is given by a smooth submanifold M in R2, and (a, b) is a point of this submanifold such that the tangent space there is not vertical (that is ), then M in some small enough neighbourhood of (a, b) is given by a parametrization (x, f(x)) where f is a smooth function. In less technical language, implicit functions exist and can be differentiated, unless the tangent to the supposed graph would be vertical. In the standard case where we are given an equation
the condition on R can be checked by means of partial derivatives.
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