I have three exams today, and for Analysis, there's a possibility we'll have to recite either the Inverse Function Theorem or the Implicit Function Theorem.

Inverse Function Theorem

Suppose f is continuously differentiable over U, some subset of Rⁿ, for f: Rⁿ → Rⁿ. Also suppose that, for some a in E, f'(a) is invertible, and that b = f(a). Then:

1. There exist U, V open in Rⁿ such that a is in U, b is in V, f is one-to-one on U, and f(U) = V
2. If g = f^-1(V) = f^-1(f(U)), then g is continuously differentiable over V

Implicit Function Theorem

Suppose that for f:R^n + m → Rⁿ, and for E an open subset of R^n + m, f is continuously differentiable over E. Suppose also that for some (a, b) in E, f(a, b) = 0. Let A = f'(a, b) = (A_x A_y), and assume that A_x is invertible.

Then there exist U in R^n + m, and W in R^m, both of which are open, so that (a, b) is in R^n + m, and b is in R^m. In fact, for every y in W, there exists a unique x such that f(x, y) = 0.

If we let this x = g(y), then f(g(y), y) = 0, g:W → Rⁿ is continuously differentiable, and g'(b) = -A_x^-1 · A_y.