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When were the standard bump function examples such as $e^{-1/(1-x^2)}$ first understood, and what was the context or motivation at the time?

As an upper bound I would guess that they must have been understood by the time of Émile Borel's 1895 Sur quelques points de la théorie des fonctions, since Hörmander cites that article as proving Borel's lemma. However, I am unable to read French to see if Borel introduces it or is developing prior work.

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    $\begingroup$ This seems like a question for HSM. $\endgroup$
    – LSpice
    Commented May 6 at 5:29
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    $\begingroup$ What's the intended purpose of the history-overview tag if not for questions like this? $\endgroup$
    – Quarto Bendir
    Commented May 6 at 16:15
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    $\begingroup$ @QuartoBendir not arguing about the appropriateness of this question one way or the other, but the tags with the two letters and a dot at the start simply match the arXiv math subject areas $\endgroup$
    – Sam Hopkins
    Commented May 8 at 13:35

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