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# Real analysis: test 4

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Jump to navigationJump to searchThis is a test for Real analysis: course.

- State the extension of the Mean Value Theorem to functions $f:\mathbf{R}^{n}\rightarrow\mathbf{R}.$
- Give example of such a function $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$ that $f$ is not continuous at $(0,0)$ but both partial derivatives exist at $(0,0).$
- State and prove the Contraction Principle. Give examples of functions for which the theorem does or does not apply.
- Describe Newton's method. Give an example of a function for which the method does not apply.
- Let $S$ be a complete metric space. Then every subset $A$ of $S$ is also a metric space. When is and when is not $A$ a
*complete*metric space? - Give examples of functions $f:\mathbf{R}\rightarrow\mathbf{R}$ that satisfy and don't satisfy the Lipschitz condition.