Geodesic
Automorphisms of the semigroup of all differentiable functions
Glasgow mathematical journal, Tome 8 (1967) no. 1, pp. 63-66

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Let R denote the space of real numbers and let D(R) denote the family of all functions mapping R into R that are (finitely) differentiable at each point of R. Since the composition f o g of two differentiable functions is also differentiable and since the composition operation is associative, it follows that D(R) is a semigroup with this operation. Such semigroups have been studied previously. Nadler, in [4], has shown that the semigroup of al differentiable functions mapping the closed unit interval into itself has no idempotent elements other than the identity function and the constant functions. The proof of that result carries over easily to the semigroup D(R). 
Jr, Kenneth D. Magill. Automorphisms of the semigroup of all differentiable functions. Glasgow mathematical journal, Tome 8 (1967) no. 1, pp. 63-66. doi: 10.1017/S0017089500000100
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[1] 1.Clifford, A. H. and Preston, G. B., The algebraic theory of semigroups, Amer. Math. Soc. Math. Surveys, No. 7, (1961). Google Scholar | DOI

[2] 2.Ljapin, E. S., Semigroups, Translations of Mathematical Monographs, Vol. 3, Amer. Math. Soc., (1963). Google Scholar | DOI

[3] 3.Magill, K. D. Jr, Semigroups of continuous functions, Amer. Math. Monthly 71 (1964), 984–988. Google Scholar | DOI

[4] 4.Nadler, S. B. Jr, The idempotents of a semigroup, Amer. Math. Monthly 70 (1963), 996–997 Google Scholar | DOI

[5] 5.Natanson, I. P., Theory of functions of a real variable, Vol. I, Frederick Ungar Publishing Co. (New York, 1955). Google Scholar

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