You cannot solve every equation for y, but you can often prove y exists
The equation x squared plus y squared equals 1 draws a circle, and no single formula y = g(x) can trace the whole thing, since most x values match two points. The implicit function theorem explains when a tangled equation still hides a well-behaved function, even one nobody can write down.
Take the circle again. Its top half is the graph of the square root of one minus x squared, and its bottom half is the negative of that expression, each defined for x from minus 1 to 1. Neither covers the full curve, but near almost any chosen point, one of them describes the circle perfectly. The theorem generalises that observation: under a mild condition, the set of solutions of an equation looks locally like the graph of a function.
The condition concerns partial derivatives. For a smooth function f of x and y, pick a point on the curve where f equals zero. If the rate of change of f with respect to y is not zero there, then in some neighbourhood y can be treated as a differentiable function of x. Differentiating the equation with the chain rule even hands you the slope directly: it equals minus the x-derivative of f divided by its y-derivative. One proof turns that slope formula into an ordinary differential equation and invokes the Peano existence theorem to guarantee a solution.
The idea scales up. Given m equations in n plus m unknowns, the m dependent variables can be expressed locally as differentiable functions of the other n whenever a corresponding condition on their derivatives holds. Those functions usually lack any closed-form expression, which is exactly why they are called implicit. The theorem remains useful even when no formula for f itself is at hand.
The first rigorous version is credited to Augustin-Louis Cauchy, who lived from 1789 to 1857. Ulisse Dini, born in 1845 and active until his death in 1918, later extended the real-variable form to functions of any number of real variables, giving the result the reach that makes it a workhorse of multivariable calculus.
Source: Implicit function theorem
