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I adore counterexamples. They're one of the most beautiful things about math: a clevery found ugly counterexample to a plausible claim. Below I have listed 10 statements about basic analysis which are all false. Your job is to find the correct counterexample. Some are easy, some are not so easy. Here are the rules:

I will give the source of each statement after the correct answer has been given.

- 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.

- SOLVED BY Samy_A Any open set ##G## such that ##\mathbb{Q}\subseteq G\subseteq \mathbb{R}## has either ##G=\mathbb{Q}## or ##G=\mathbb{R}##.
- SOLVED BY ResrupRL Every symmetric matrix in ##\mathcal{M}_n(\mathbb{C})## is diagonalizable for each ##n##
- SOLVED BY stevendaryl Every ##C^1## function ##f:\mathbb{R}\rightarrow \mathbb{R}## that is square-integrable, that is ##\int_{-\infty}^{+\infty} |f(x)|^2 dx <+\infty##, has ##\lim_{x\rightarrow +\infty} f(x) = 0##.
- SOLVED BY andrewkirk There is no infinitely differentiable function ##f:\mathbb{R}\rightarrow \mathbb{R}## such that ##f(x) = 0## if and only if ##x\in \{1/n~\vert~n\in \mathbb{N}\}\cup\{0\}##.
- SOLVED BY ProfuselyQuarky Every derivative of a differentiable function is continuous.
- SOLVED BY Samy_A For any ##A\subseteq \mathbb{R}## open holds that if ##f^\prime(x) = g^\prime(x)## for each ##x\in A##, then there is some ##C\in \mathbb{R}## such that ##f(x) = g(x) + C##.
- SOLVED BY fresh_42 If ##(a_n)_n## is a sequence such that for each positive integer ##p## holds that ##\lim_{n\rightarrow +\infty} a_{n+p} - a_n = 0##, then ##a_n## converges.
- SOLVED BY jbriggs444 There is no function ##f:\mathbb{R}\rightarrow \mathbb{R}## whose graph is dense in ##\mathbb{R}^2##.
- SOLVED BY andrewkirk If ##f:\mathbb{R}^2\rightarrow \mathbb{R}## is a function such that ##\lim_{(x,y)\rightarrow (0,0)} f(x,y)## exists and is finite, then both ##\lim_{x\rightarrow 0}\lim_{y\rightarrow 0} f(x,y)## and ##\lim_{y\rightarrow 0}\lim_{x\rightarrow 0} f(x,y)## exist and are finite.
- SOLVED BY andrewkirk If ##A## and ##B## are connected subsets of ##[0,1]\times [0,1]## such that ##(0,0),(1,1)\in A## and ##(1,0), (0,1)\in B##, then ##A## and ##B## intersect.

I will give the source of each statement after the correct answer has been given.

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