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# Real analysis: test 2

This 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 $S$ ia also a metric space. Whan 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.

1. State the Heine-Borel Theorem and provide two examples that show that the conditions of the theorem cannot be removed.

2. (a) State Rolle's Theorem and the Mean Value Theorem. (b) Prove one of them.

3. Discuss the differentiability of the functions $$f(x)=\left\{ \begin{array} [{}% x\sin\frac{1}{x} & \text{if }x\neq0\\ 0 & \text{if }x=0 \end{array} \right. $$ and $$g(x)=\left\{ \begin{array} [{}% x^{2}\sin\frac{1}{x} & \text{if }x\neq0\\ 0 & \text{if }x=0 \end{array} \right. $$

4. From the definition of Darboux integral, prove that $\int_{a}^{b}cdx=c(b-a),$ where $c$ is a constant.

5. State and prove the Fundamental Theorem of Calculus.

6. Is it possible that $|f|$ is integrable on $[0,1]$ but $f$ is not?

7. #Test for convergence $$\sum\frac{2n+1}{n^{3}}.$$