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# Category Archives: Analysis

## Implicit function theorem

Implicit funtion theorem states that if (or nonsingular), then near we can uniquely solve from the equation . Moreover smoothness of determines smoothness of . Usually at least is . A deep result is that in order to solve a continuous … Continue reading

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## Fubini-Tonelli principle

Fubini-Tonelli principle: Imagine that you have a saw-shaped “square” with very deep sawteeth and small area. If we integrate iteratedly, then in one order, in the inner integral we can only bound with and thus the estimate would be , … Continue reading

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