Abstract. The well-known Takagi function T(x)plays a crucial role in the theory of approximately convex functions. In order to establish the sharpness of some Bernstein-Doetsch type results for approximate convexity, we prove that the Takagi function fulfils the inequalityfor all real numbers x and y .Mathematics subject classification (2000): 26A51, 26A30, 39B62.

In this paper the following implication is verified for certain basic algebraic curves: if the additive real function f approximately (i.e., with a bounded error) satisfies the derivation rule along the graph of the algebraic curve in consideration, then f can be represented as the sum of a derivation and a linear function. When, instead of the additivity of f , it is assumed that, in addition, the Cauchy difference of f is bounded, a stability theorem is obtained for such characterizations of derivations.

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