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Well, the last thread of counterexamples was pretty fun. So why not do it again! Again, I present you a list with 10 mathematical statements. The only rub now is that only ##9## are false, thus

Rules:

Here you go:

Thank you all for participating! I hope some of these statements were surprising to some of you and I hope some of you have fun with this! Don't hesitate to post any feedback in the thread!

**one of the statements is true**. Provide a counterexample to the false statements and a proof for the correct one!Rules:

- For a counterexample to count, the answer must not only be correct, but a detailed argumentation must also be given as to why it is a counterexample.
- Any use of outside sources is allowed, but do not look up the question directly. For example, it is ok to go check analysis books, but it is not allowed to google the exact question.
- If you previously encountered this statement and remember the solution, then you cannot participate in this particular statement.
- All mathematical methods are allowed.

Here you go:

- SOLVED BY PeroK ##\mathbb{R}## is the disjoint countable union of closed intervals
- SOLVED BY mfb Any open set in ##\mathbb{R}^2## is the disjoint countable union of open balls, where an open ball is a set of the form ##B(\mathbf{a},r) = \{\mathbf{x}\in \mathbb{R}^2~\vert~(a_1 - x_1)^2 + (a_2 - x_2)^2 < r^2\}## for ##\mathbf{a}\in \mathbb{R}^2## and ##r>0##.
- SOLVED BY andrewkirk Let ##\pi:\mathbb{N}\times \mathbb{N}\rightarrow \mathbb{N}## be a bijection and let ##\sum_{n} a_n## be an absolutely convergent series in ##\mathbb{R}##. Then ##\sum_n a_n = \sum_{k=0}^\infty \sum_{l=0}^\infty a_{\pi(k,l)}##.
- SOLVED BY PeroK Every continuous bounded function ##\mathbb{R}\rightarrow \mathbb{R}## is uniformly continuous.
- SOLVED BY Samy_A Every function ##\mathbb{R}\rightarrow \mathbb{R}## is the derivative of some function.
- SOLVED BY PeroK If ##f:\mathbb{R}\rightarrow \mathbb{R}## is continuous and sends open sets to open sets, then it also sends closed sets to closed sets.
- SOLVED BY mfbThere is no monotonic function ##\mathbb{R}\rightarrow \mathbb{R}## whose set of discontinuities is ##\mathbb{Q}##.
- SOLVED BY Samy_A Every nonconstant function ##\mathbb{R}\rightarrow \mathbb{R}## that is periodic has a smallest period.
- SOLVED BY andrewkirk For any infinitely differentiable monotonic function ##f:\mathbb{R}\rightarrow \mathbb{R}## such that ##\lim_{x\rightarrow +\infty} f(x) = 0##, also holds that ##\lim_{x\rightarrow +\infty} f^\prime(x) = 0##.
- SOLVED BY andrewkirk Every subset of ##[0,1]## of measure zero is the set of points of discontinuity of a Riemann-integrable function ##[0,1]\rightarrow \mathbb{R}##.

Thank you all for participating! I hope some of these statements were surprising to some of you and I hope some of you have fun with this! Don't hesitate to post any feedback in the thread!

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