the necessary and sufficient condition for the integrability
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The necessary and sufficient conditions for complete integrability of a Pfaffian system are given by the Frobenius theorem. One version states that if the ideal then the system admits a foliation by maximal integral manifolds. (The converse is obvious from the definitions.) Not every Pfaffian system is completely integrable in the Frobenius sense.
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Every continuous function [a,b] →ℝ is uniformly continuous. There are three natural proofs of this theorem, using (a) The Nested Interval Theorem, (b) the Heine-Borel Theorem, and (c) the Bolzano-Weierstrass Theorem respectively.
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