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# Real analysis: test 1

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

- Give two non-Euclidean metrics on $\mathbf{R}^{2}.$ Prove.
- Prove that an open ball in a metric space is an open set.
- Prove that a compact set in a metric space is bounded and closed.
- Suppose $S,T$ are metric spaces and $f,g:S\rightarrow T$ are continuous functions. Prove that the set $A=\{x\in S:f(x)=g(x)\}$ is closed in $S.$ What can you say about $B=\{x\in S:f(x)\neq g(x)\}?$
- State and prove the fundamental lemma of differentiation for $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$.
- State the definition of a differentiable function $f:\mathbf{R}^{N}\rightarrow\mathbf{R.}$ Give an example of a function $f:\mathbf{R}^{2}\rightarrow\mathbf{R}$ such that both partial derivatives of $f$ exist at $x=a,$ but $f$ is not differentiable.

- Use the Sandwich Theorem to prove that $\lim\limits_{x\rightarrow0}x^{n}=0$ for every positive integer $n.$
- State the definition of the one-sided limit and state the basic theorems about it.
- From the definition, show that $f(x)=x^{2}-1$ is uniformly continuous on $[0,1].$
- State the Nested Intervals Theorem and provide examples that show that the conditions of the theorem cannot be removed.
- State and prove the theorem about boundedness of continuous functions.
- Prove that any subsequence of a convergent sequence converges.