Is it possible to extend a function in Smooth Infinitesimal Analysis? Is it possible in general in intuitionistic mathematics?

Question

Assume that in the standard setting I have a (continuous) function $f:\mathbb{R\setminus\{0\}}\to\mathbb{R}$, with limit at $0$ equal to L. E.g. $f(x)=\frac{sin(x)}{x}$. Then I can extend $f$ to (continuous) $$ g(x)= \begin{cases} f(x),\, x\neq 0 \\ L,\, x=0 \\ \end{cases} $$

Now, few questions:

1. Can $f$ even be defined? My understanding is $\mathbb{R}$ cannot be detached into $\mathbb{R}\setminus\{0\}$ and $\{0\}$. Am I correct? Can this be remedied somehow?
2. Is the above non-detachability rule happens only in SIA, or in intuitionistic mathematics in general?
3. What about $g$ being defined by cases? Even if it gives raise to a continuous function, there will be some $x$ that we cannot decide if they are equal to $0$ or not, thus we won't be able to apply $f$ over them? This would be a problem to intuitionism in general?
4. Existence of L is given, but in general how do we define a limit in SIA?
5. In the case of $f(x)=\frac{sin(x)}{x}$ I can define $g$ using Taylor series? That is since $$ \frac{sin(x)}{x}=\frac{x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \cdots}{x} $$ we can define $g$ as $$ g(x) = 1 - \frac{x^2}{3!} + \frac{x^4}{5!} - \frac{x^6}{7!} + \cdots $$ Is this correct approach in SIA? Do we have a global Taylor series available, or only finite versions for nilsquares, nilcubes, etc.?
6. How do we define sine function in SIA? I saw John L. Bell doing it in a geometric way, but can we do it via Taylor series?

Answer 1

In what follows, let $R$ denote the real line object of Smooth Infinitesimal Analysis (SIA).

You can of course define the function $f : R\setminus \{0\} \rightarrow R$. The domain of this function is $R\setminus \{0\}$, i.e. $\{ x \in R \:|\: x \neq 0 \}$.

The function $y \mapsto \frac{1}{y}$ is defined on the object $R \setminus \{0\}$ by the standard axioms characterizing $R$ in SIA. Similarly, you have functions $y \mapsto \sin(y)$ and $(x,y) \mapsto x\cdot y$ defined on all of $R$, so the composition $y \mapsto \sin(y) \cdot \frac{1}{y}$ defines a function of signature $R \setminus \{0\} \rightarrow R$ as expected.

The fact that you cannot write $R$ as $R \setminus \{0\} \cup \{0\}$, while true, is not a particularly relevant consideration here.

2.

No, if $\mathbb{R}$ denotes the set of (Dedekind or Cauchy) reals, you cannot prove that $\mathbb{R}$ coincides with $\mathbb{R}\setminus \{0\} \cup \{0\}$ in any intuitionistic or constructive set theory.

3.

You can define the function $g$ by cases as you do above. The domain of the function $g$ defined this way will be $R\setminus \{0\} \cup \{0\}$, however. What does this mean? It means you can apply $g$ to:

• any $x \in R \setminus \{0\}$, i.e. any $x\in R$ for which $x \neq 0$ holds; and
• any $x \in \{0\}$, i.e. any $x \in R$ for which $x = 0$ holds (we have a unique such $x$).

However, you cannot apply $g$ to an $x$ for which you cannot establish either of the above. In particular, you cannot write $g(\varepsilon)$ for some nilsquare infinitesimal $\varepsilon^2 = 0$, unless you also know that $\varepsilon = 0$ holds.

4,5,6.

Normally, when we work in smooth infinitesimal analysis, we assume that we have access to all smooth functions on $\mathbb{R}$ as functions $f: R \rightarrow R$: this amounts to saying that $R$ constitutes a smooth algebra. You don't ever have to worry about whether some smooth function is definable in SIA: the answer is always yes, pretty much by definition, and these smooth functions work on infinitesimals exactly the way that the respective Maclaurin series suggest.

Of course, if you want to use Smooth Infinitesimal Analysis as a sort of pure axiomatic "foundation" for calculus, this approach does not work: figuring out which functions are the smooth ones is part of the work that a foundation should let us do. But even if you use SIA this way, you still have the integration axioms and the implicit function theorem, which allows constructing many of the familiar smooth functions one would want on $R$.

The question of limits is similarly complicated. Different methods apply depending on what your goal is, and some approaches involve working with models of Smooth Infinitesimal Analysis containing both nilpotent infinitesimals and invertible infinitesimals. And since most users of SDG don't often need limits for any purpose, it's also necessarily a niche topic. While it might work as a separate question, I suggest coming back to it after working through one or more textbooks on Smooth Infinitesimal Analysis or Synthetic Differential Geometry. This would preferably include Moerdijk-Reyes, but fully understanding how Bell does things (without limits) would also provide good context.

Answer 2

Note that I am not familiar with SIA, so the following answer will be for constructive mathematics in general with $\mathbb{R}$ being the constructive reals. Here, I will view a constructive real $\alpha$ roughly as a function which takes a natural number $n$ and returns a rational number which is meant to represent an approximation to $\alpha$ to within say $\frac{1}{n}$. (In particular, this situation does not really have any way to distinguish infinitesimals from 0, so it might not be suitable for SIA as far as I know.)

You are correct that $f$ is not necessarily defined on $\mathbb{R} \setminus \{ 0 \}$. However, something close is valid: $f$ is defined on the set of real numbers which are separated from 0 -- where a constructive real is separated from 0 if there is some $n$ such that the approximation of $\alpha$ to within $\frac{1}{n}$ is either greater than $\frac{2}{n}$ or less than $-\frac{2}{n}$, so the approximation is sufficiently far from 0 that it ensures the real number cannot be zero. Given such a number separated from 0, you can find a positive rational number which is a lower bound for $|\alpha|$, and using that lower bound you can build an algorithm for using approximations to $\alpha$ to find sufficiently accurate approximations to $\frac{1}{\alpha}$.

Now to define $g$, the rough idea on how to do that is as follows: first, based on your desired accuracy $\frac{1}{n}$ for $g$, choose some $m$ and take an approximation to $\alpha$ accurate to within $\frac{1}{m}$. Now, if that approximation establishes that $\alpha$ is separated from 0, then use that (possibly calling the function again with larger values of $m$) to calculate the desired approximation to $\sin(\alpha) \cdot \frac{1}{\alpha}$. Otherwise, you simply return 1 as the approximation to $g(\alpha)$.

Note that the hypothesis $\lim_{x\to 0} \frac{\sin x}{x} = 1$ is essential to be able to establish that the result gives a well-defined constructive real -- in other words, that the sequence of approximations you return satisfies a required coherence condition. For example, if $x$ is close to $0.01$, then for $n$ up to something on the order of $10^4$, you can return an approximation of 1. For larger $n$, though, you will detect the difference of $x$ from 0, and start using the $\sin(x) \cdot \frac{1}{x}$ algorithm to get better approximations; however, you do need to know that those better approximations are still sufficiently close to 1 to agree with the earlier crude approximations.