Mathematics
The Implicit Function Theorem and Its Applications in Calculus
Quick fact
The implicit function theorem guarantees that even when you cannot solve an equation like x^2 + y^2 = 1 for y as a single formula, there exists a local function y = f(x) near almost every point on the curve.
Why this is interesting
Have you ever wondered how we can describe a curve like a circle as a function, even though it's not a function of x? The implicit function theorem holds the answer.