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The phrase "a third function" is correct and usable in written English.
It can be used when referring to an additional role or purpose in a context where multiple functions are being discussed.
Example: "In addition to its primary and secondary functions, the device also has a third function that enhances its usability."
Alternatives: "an additional function" or "a tertiary function".
Exact(10)
A third function is to reduce the temperature of the base metal in the case of thermal barrier coatings.
A third function is to reduce the temperature of the base metal in the case of thermal barrier applications.
A third function of worship is that of providing personal support to individual members of a group or of a society.
A third function of doctrine is polemical: the defense of the faith against misinterpretation and error, within or without a religious tradition.
Aaron J. Corcoran and William E. Conner of Wake Forest University, and Jesse R. Barber of Colorado State University, show that for the Bertholdia trigona species, the clicks serve a third function.
A third function or capacity of "the knowing" is discursive knowledge, here equated with understanding (A6).
Similar(50)
Under the process of "addition exaptation", a feature that is functional in one capacity assumes a second function without losing its original function (Arnold 1994).
But its name suggests a second function.
Throughout the apartheid period, art centers served a second function.
Eyelashes have, like many other bodily features, acquired a second function as a signal.
In short order, Roux took on a second function as Guinness' controller; later he became the financial director.
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com