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Consider the set, let's call it $R$, of all functions $f:\mathbb{R} \to \mathbb{R}$? Are there any interesting subsets $S \subset R$ whose cardinality is (currently) unknown? By interesting, I mean something "natural" (e.g., continuous real functions, the set of all real functions with a countable number of discontinuities, Darboux functions) which analysts might care about.

The motivation for this question is honestly just curiosity. I find this quite interesting, because there's so many types of real functions we can consider. Surely there's some interesting, nontrivial examples worth discussing.

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How about the space of all real-valued functions which are not continuous? You can actually show that there are some input-output pairs of real numbers which satisfy the definition of continuity. In other words, you can show continuous functions exist without ever writing one down!

Proof $(1)$ Let $F(\mathbb{R},\mathbb{R})$ denote the set of all possible single variable, real valued functions.

$(2)$ Now let $S\in F(\mathbb{R},\mathbb{R}) $ denote the set of all functions where if the difference between output values is less than $\epsilon$, then we can find a corresponding $\delta$ which will always be greater than the difference between two arbitrary input values. This is possible, we can just make up a rule where the difference between input values $x_1$ and $x_2$ is twice the difference between $f(x_1),f(x_2)$. Play around with some numbers to get the intuition.

$(3)$ Let $f\in S$. Consider two points of the function $f(x_{1})$ and $f(x_{2})$. Remember since $f\in S$, the difference $|f(x_{1})-f(x_{2})|<\epsilon$ and $|x_1-x_2|<2\delta$ by definition. This fits the definition of continuity, and so continuous functions do exist. You can modify to show its possible to have differentiable functions, monotonic etc.

What is the cardinality of $F(\mathbb{R},\mathbb{R})$? What about the set of monotone functions? What about the set of functions which are differentiable, but the derivative is not differentiable anywhere?

To conclude, you could reasonably conjecture (based on cardinality of power sets) that there are probably a $\aleph_{2}$ number of possible subsets of $F(\mathbb{R},\mathbb{R})$ we could ask ourselves what the cardinality of each of those sets are. Consider a set $X$ where each function is bounded by some constant: $|f(x)|<M$. We could choose any $M\in (1,\infty)$ to be the bound, and that means there are $\aleph_1$ such functions in $X$ alone.

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  • $\begingroup$ OP asked for an interesting set of functions whose cardinality is unknown. How does this answer address that? $\endgroup$ – Ittay Weiss Apr 12 '17 at 5:43
  • $\begingroup$ it was clearly stated. How many monotone functions, how many bounded functions etc. $\endgroup$ – Kernel_Dirichlet Apr 12 '17 at 5:58
  • $\begingroup$ What makes you think these are unknown cardinalities? $\endgroup$ – Ittay Weiss Apr 12 '17 at 6:50
  • $\begingroup$ "What about the set of monotone functions? What about the set of functions which are differentiable, but the derivative is not differentiable anywhere?" These are all Baire functions (of the simplest types) and there are $c$ many Baire functions, so there are at most $c$ many of each of these types. There are at least $c$ many of each of these types because any nonzero constant added to a specific function belonging one of these types produces a different function of the same type. $\endgroup$ – Dave L. Renfro Apr 12 '17 at 13:46
  • $\begingroup$ well then I learned something as well. You could have spent your time answering the question rather than downvoting mine for no reason other than I was unaware that these are Baire functions. $\endgroup$ – Kernel_Dirichlet Apr 12 '17 at 15:33

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