## On derivations on certain function spaces and their relation to the differential operator. (Über Derivationen auf gewissen Funktionenräumen und deren Beziehung zum Differentiationsoperator.)(German)Zbl 0813.39008

The origin of this paper was, as stated, a talk by the first author aimed at characterizing derivatives on the set $$C$$ of continuous functions on a real interval $$I$$ by a system of functional equations and a contribution by the second author to the discussion, showing that only the identically zero operator satisfies these equations on $$C$$.
Here they prove, among others, that if the domain of an operator is an algebra which contains $$C$$ and is contained in the set of real valued functions on $$I$$ and if it is a derivation (i.e. satisfies the rules for derivatives of sums and products) then it maps every differentiable function on $$I$$ onto the product of its derivative and of a constant, which is the value of the operator on the identity function.

### MSC:

 39B52 Functional equations for functions with more general domains and/or ranges 39B62 Functional inequalities, including subadditivity, convexity, etc. 46G05 Derivatives of functions in infinite-dimensional spaces 26A24 Differentiation (real functions of one variable): general theory, generalized derivatives, mean value theorems
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