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Yes. Let $A$ be the set of reals whose decimal expansion has infinitely many occurrences of 7, and let $B$ be the set of reals whose expansions have only finitely many such occurrences. These sets p …

The sets you refer to as "ultranegligible" are known as the strong measure zero sets, and the assertion that every strong measure zero set is countable is known as the Borel conjecture, and is indepen …

Although you asked about continuous functions, here is an example of a discontinuous function $f:[0,1]\to\mathbb{R}$ which is not monotone on any measurable set with positive measure. Let $V\subset …

In the case of the reals, the uncountable version reduces easily to the countable version, since there will be a countable subfamily of the given uncountable family that is cofinal in the inclusion or …

Here is an easier proof that the cardinality is continuum. (Perhaps this is what Gowers was suggesting.) First, any Cauchy sequence is equivalent to any of its subsequences. Second, if the original s …

For any values of $L_1$, $L_2$ and $L_3$, whether equal or not, there is a sequence $a_{nm}$ achieving your three limit requirements. The reason is that $\lim_n a_{nn}=L_1$ refers only to the entries …

Your collection is not closed under complement. To see this, observe that the diagonal $\Delta=\{(x,x)\mid x\in\mathbb{R}\}$ is not in your collection, since the only rectangles it contains are single …

Yes, just take a copy of the ordinal $\omega^\omega$ in the reals. This has Cantor Bendixson rank exactly $\omega$. One way to see this is first to understand how to make a closed set last for exact …

If you allow the functions to be constant on some intervals, then there are some easy examples, and Ricky has provided one. But if you rule that out, then there can be no examples, even with countab …

There is no such computable decision procedure, largely because of the same issues underlying Rice's theorem, as mentioned in the comment of Steven Stadnicki. The moral of Rice's theorem is that you c …

The initial function you mention is the diagonalizing function $d:\mathbb{R}^\omega\to\mathbb{R}$, for which one ensures that $z=d(x_0,x_1,\ldots)$ is distinct from every $x_n$ simply by making the $n …

The other answers and comments are fascinating, particularly about the irrationality measure, but allow me to give a little more information along the lines of Mark Sapir's answer by mentioning that t …

Nice question, Erin. Here is one quick easy thing to say. If $\pi$ and $e$ disagree in infinitely many digits, then there are continuum many choices of the particular subset of those digits to swap, …

If the continuum hypothesis holds, then there is such an ideal. Indeed, we need only to assume that $2^\omega\lt 2^{\omega_1}$, a weakening of CH. Consider the tree $T=2^{\lt\omega_1}$ of all binary …

The distance function to a closed set is continuous, even Lipschitz continuous, and is zero exactly on that closed set. A modified version of this function can be made continuously differentiable, by …