What Are the Conditions on f and g for y′(x) to Exist?

[mistereko]

Let F(x, y) = f(y − x + g(y + x)), where f(u) and g(u) are sufficiently
differentiable functions of a single real variable. If, in a neighbourhood of
(x, y) = (a, b), the equation F(x, y) = 0 defines a function y(x), state the
condition(s) on f and g so that y′(x) exists in a neighbourhood of x = a.

Homework Equations

I've done all that maths part after this question, but I'm not sure how to define it. It uses the implicit function theorem.

The Attempt at a Solution

 

[lippyka]

For y′(x) to exist in a neighbourhood of x = a, both f and g must be continuous on the interval (a - b + g(a + b), a + b + g(a - b)) and both the partial derivatives ∂F/∂y and ∂F/∂x must be non-zero at (a, b).