I have two related questions:
- Who first defined the oscillation function (perhaps under a different name)?
- When did the switch from the phrase "saltus function"(*) to "oscillation function" happened?
Recall that given a topological space $X$, a metric space $M$, and a function $f:X\to M$, the oscillation function of $f$ is defined as:
For what it's worth, Froda, in his 1929 thesis * Sur la distribution des propriétés de voisinage des fonctions de variables réelles * (https://eudml.org/doc/192780) attributes (p.23) the observation that the higher order oscillations stabilize to a 1905 paper by Denjoy (where this is used implicitly, Froda says as far as I can see) and a 1910 paper by Sierpinski.
(There are other, non-pointwise definitions, or else definitions in terms of the difference between limit superiors and limit inferiors also; these definitions are also admissible for the purposes of this question.)
(*): See e.g. Hobson's 1921 book The theory of Functions of a Real Variable and the Theory of Fourier's Series, Vol. I (2e) , p.291, or Blumberg's 1917 paper "Certain General Properties of Functions".
Finally, here is the OED definition of "saltus" (I hadn't heard of this word before):
In light of the definition of the word it seems more appropriate to call the function at hand the saltus function than the oscillation function (also keeping in mind the more commonly studied "BMO functions").