Jordan's curve theorem is used as a blackbox.
This topology theorem states that a looped continuous path in the plane partitions the points of the plane, such that any continuous path going from a point in one partition to a point in the other intersects the loop.
There seem to be a lot of theorems in calculus of which I don't fully understand the proof, though some of this shows my ignorance of calculus. Jordan's theorem seem to be an extreme example though. Let me list some other examples.
- the existance and basic properties of the Lebesgue measure and infinite product measures
- the fact that a Wiener process is almost surely everywhere continuous (mentioned below as a separate answer by weakstar)
- the fact that the roots of a complex polynomial (or the eigenvalues of a complex matrix) are continuous in the coefficients (though I should learn the proof for this because the more precise statements on how well conditioned the roots are on the coefficients is useful)
- the spectral theorem about linear maps on a possibly infinite-dimensional Hilbert-space
the proof that a convex function (from reals to reals) is always continuous everywhere and has a left and right derivative everywhere(Update: okay, remove this last one because Ian Morris gave a simple proof below. I seemed to remember it was more difficult than that. Thanks, Ian.)- Rademacher's theorem: every Lipschitz function from an open subset of $ \mathbb{R}^m $ to $ \mathbb{R}^n $ is differentiable almost everywhere. (Added on Paul Siegel's suggestion. For some reason I haven't heared of this theorem before, but it sure sounds useful.)
- Lebesgue's criterium which claims that a bounded function from reals to reals is Riemann-integrable iff it's continuous almost everywhere. (The proof is elementary and doesn't require any ideas, but it's laborous.)