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I am looking for a list of mathematical technicalities that are not so well-known, even in the mathematical community. What I mean is, I am looking for examples of phenomenon where it is important to know the formal definitions, where it is important that the hypotheses of the theorem are satisfied, rather than "almost-satisfied". Let me give a simple example. There is a "theorem" that most mathematicians believe, which is that anti-derivatives of a function $f$ differ by a constant. However, that "theorem" is false as stated without the hypothesis that the function is over a connected domain. For example, given a continuous function $f$ on the domain $\mathbb{R} - \{0\}$, the set of anti-derivatives of $f$ is a two-dimensional set, parametrized by an ordered pair of real numbers $(x,y)$. I hope this example has made it clear of the kinds of thing I am looking for.

user107952
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    Does this answer your question? Examples of common false beliefs in mathematics – Sam Hopkins Jun 07 '22 at 01:39
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    " I am looking for examples of phenomenon where it is important to know the formal definitions, where it is important that the hypotheses of the theorem are satisfied". Have you tried opening a randomly chosen math book to a randomly chosen page? – Steven Landsburg Jun 07 '22 at 03:24
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    The set of antiderivatives of $x\mapsto\sec^2 x$ for $x\in\mathbb R$ is $(\tan x + \text{piecewise constant})$ where the $\text{“piecewise constant”}$ is constant on each interval $\left( -\pi/2 + n\pi, +\pi/2 + n \pi \right), ,,, n\in\mathbb Z.$ But a widely used calculus textbook says it is $x\mapsto(\tan x+\text{constant}). \qquad$ – Michael Hardy Jun 07 '22 at 05:09
  • @MichaelHardy: If there were a way to settle this wager, I would bet you a very large sum that the author of the textbook understood this perfectly well but had a momentary brain lapse while typing. – Steven Landsburg Jun 08 '22 at 01:06
  • @StevenLandsburg : I would guess that the author would instantly acknowledge this if it were pointed out, but wasn't thinking about it. – Michael Hardy Jun 09 '22 at 23:29

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