The classical form of the oscillator above is $\sin(1/x)$, which this library can only construct much later

Every analysis course states the two examples of this page in the form

$\sin(1/x) \quad \text{has no limit at } 0, \qquad x \sin(1/x) \to 0 \text{ as } x \to 0 ,$

and a reader who has met them before will recognise $\psi(1/x)$ has no limit at $0$: two sequences tending to $0$ give values constantly $0$ and constantly $1/2$ and $x\,\psi(1/x) \to 0$ as $x \to 0$, by the squeeze theorem as those examples with $\psi$ in place of $\sin$. This remark records the correspondence, and it records that the correspondence is orientation only: the two displayed statements are reported as what the classical treatment proves, not asserted here, and nothing on this page uses or proves anything about $\sin$.

This library now constructs sine and cosine from their power series, proves their differential and addition laws, and defines pi from the first positive zero of cosine. Under this library's counterexample convention, sin(1/x) has no limit as x tends to zero ↗ displays the false proposition that the sine limit exists under Statement refuted, then proves it false; x sin(1/x) tends to zero despite its oscillation ↗ proves the squeezed limit for the product. Both occur later in the reading order, so the links are orientation-only forward references declared in this item's forward_refs; no proof on this earlier page depends on them.

The function $\psi(x) = \inf_{n \in \mathbb{Z}} |x - n|$ of The trigonometry-free oscillator $\psi(x) = \inf_{n \in \mathbb{Z}} |x - n|$ is well defined and attained at a nearest integer, takes values in $[0, 1/2]$, vanishes exactly on $\mathbb{Z}$, equals $1/2$ at half-integers, and is $1$-periodic is elementary — it needs only the integer part, the order and the absolute value — and it has the three properties that make the classical examples work:

• it is bounded, with values exactly in $[0, 1/2]$;
• it is periodic, with period $1$, so $\psi(1/x)$ oscillates without damping as $x \to 0$;
• it attains two distinct values on every punctured neighbourhood of $0$ after the substitution $x \mapsto 1/x$, namely $0$ at the reciprocals of the integers and $1/2$ at the reciprocals of the half-integers.

The third property is what $\psi(1/x)$ has no limit at $0$: two sequences tending to $0$ give values constantly $0$ and constantly $1/2$ uses. It is not sharper than what $\sin$ would give — the classical witnessing sequences hit the extreme values of $\sin$ exactly too — but it is available here: the two values $0$ and $1/2$ are read off from the integer part in one line (Integer part: for every real $x$ there is exactly one integer $m$ with $m \le x < m + 1$), with no series and no $\pi$, whereas the corresponding facts about $\sin$ presuppose the whole construction described above.

Nothing on this page is weaker for using $\psi$. The two statements proved are exactly the statements usually proved with $\sin$, and their proofs are shorter.

What is lost is a connection to a different subject. The classical pair $\sin(1/x)$, $x\sin(1/x)$ also carries information about smoothness, about power series and about the topologist's sine curve, none of which $\psi$ can carry, since $\psi$ is assembled from the order, the absolute value and the integer part alone. Those notions occur only later in the reading order and are unavailable on this earlier page.