Real analysis: test 4

Grading

This is a test for Real analysis: course.

1. State the extension of the Mean Value Theorem to functions $f:\mathbf{R}^{n}\rightarrow\mathbf{R}.$
2. 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 $(0,0).$
3. State and prove the Contraction Principle. Give examples of functions for which the theorem does or does not apply.
4. Describe Newton's method. Give an example of a function for which the method does not apply.
5. 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?
6. Give examples of functions $f:\mathbf{R}\rightarrow\mathbf{R}$ that satisfy and don't satisfy the Lipschitz condition.