In thermodynamics, Carnot function (TR:9), symbol C, is the following function:

$C = \frac{1}{T}\,$

namely the inverse of the absolute temperature T of the body. [1]

Overview
In 1834, Emile Clapeyron, in his “Memoir on the Motive Power of Heat”, in his discussion of the work of Sadi Carnot (1824), supposedly (Cardwell, 1971), alluded, defined, and or outlined the roots of the Carnot function in some way; the following is a take on this: [4]

In 1849, William Thomson, in his “An Account of Carnot’s Theory of the Motive Power of Heat: with Numerical Results Deduced from Regnault’s Experiments on Steam”, is said (Cardwell, 1971) to have defined things such that μ (mu) was the reciprocal of Carnot’s function C: [5]

In 1850, Rudolf Clausius, in his 1850 paper “On the Motive Power of Heat”, building on precursory formulations of William Thomson (1849), Emile Clapeyron (1834), and Sadi Carnot (1824), is said to have derived the Carnot function exactly. [2] Specifically, Carnot had initially shown that the efficiency e of a Carnot engine working in the infinitesimal temperature range dt at temperature t is:

$e = F(t) dt\,$

where F(t) is a universal function of temperature t. [3] Building on this, Clausius showed that:

$F(t) = \frac{1}{{273 + t}} = \frac{1}{T} \,$

with t in degrees Celsius.

In 1865, Clausius, in the footnotes to Memoir One (1850), of his textbook of memoirs, following some derivation referred C as the Carnot function, which he was it was "frequently called", in the context of the following formula: [6]

Quotes
The following are related quotes:

“The reciprocal of Carnot’s function, as defined by Clapeyron, is the work done by a unit of heat ‘falling through’ one degree, and on the gas scale it decreases as the temperature increase.”
Donald Cardwell (1971), From Watt to Clausius (pg. 239) [4]

Thermodynamic function

References
1. Alekseev, G.N. (1978). Energy and Entropy (translator: U.M. Taube) (IA). Moscow: Mir Publishers.
2. Clausius, Rudolf. (1850). “On the Moving Force of Heat and the Laws of Heat which may be Deduced Therefrom”, Communicated to the Academy of Berlin, Feb.; in: Poggendorff’s Annalen, March-April, Vol. lxxix, pgs. 368, 500; English translation in: Philosophical Magazine, July 1851, Vol. ii. pgs. 1, 102. Publisher.
3. Müller, Ingo. (2007). A History of Thermodynamics: the Doctrine of Energy and Entropy (Carnot function, pgs. 54-61). New York: Springer.
4. (a) Clapeyron, Émile. (1834). “Memoir on the Motive Power of Heat”, Journal de l’Ecole Polytechnique. XIV, 153 (and Poggendorff's Annalender Physick, LIX, [1843] 446, 566). Publisher.
(b) Clapeyron, Emile. (1837). “Memoir on the Motive Power of Fire” (pg. 347-58), Taylor’s Scientific Memoirs, Volume One.
(c) Cardwell, Donald S.L. (1971). From Watt to Clausius: the Rise of Thermodynamics in the Early Industrial Age (pg. 239). Cornell University Press.
5. (a) Thomson, William. (1849). “An Account of Carnot’s Theory of the Motive Power of Heat – with Numerical Results Deduced from Regnault’s Experiments on Steam”, (127-203) Transactions of the Edinburgh Royal Society, xiv.; Annales de Chime, xxxv. 1852.
(b)
Cardwell, Donald S.L. (1971). From Watt to Clausius: the Rise of Thermodynamics in the Early Industrial Age (pg. 242). Cornell University Press.
6. Clausius, Rudolf. (1865). The Mechanical Theory of Heat: with its Applications to the Steam Engine and to Physical Properties of Bodies (translator: Thomas Hirst) (Ѻ) (Carnot's function, pg. 46). John van Voorst, 1867.