### A sandwich with convexity.

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We establish conditions under which Baire measurable solutions f of $$\Gamma (x,y,|f\left(x\right)-f\left(y\right)\left|\right)=\Phi (x,y,f(x+{\phi}_{1}\left(y\right)),...,f(x+{\phi}_{N}\left(y\right)))$$ defined on a metrizable topological group are continuous at zero.

The scientific output of Marek Kuczma consists of 179 papers published in the years 1958-1993 and three books still used and quoted. Professor Marek Kuczma created and developed the theory of iterative functional equations but his name is also connected to important results on functional equations in several variables, in particular on Cauchy's equation and Jensen's inequality. In fact Marek Kuczma has founded a mathematical school: he supervised 13 Ph.D. dissertations, 10 his students have already...

Let E be a separable real inner product space of dimension at least 2 and V be a metrizable and separable linear topological space. We show that the set of all orthogonally additive functions mapping E into V and having big graphs is dense in the space of all orthogonally additive functions from E into V with the Tychonoff topology.

We show that the solution to the orthogonal additivity problem in real inner product spaces depends continuously on the given function and provide an application of this fact.

Let $E$ be a real inner product space of dimension at least 2 and $V$ a linear topological Hausdorff space. If $cardE\le cardV$, then the set of all orthogonally additive injections mapping $E$ into $V$ is dense in the space of all orthogonally additive functions from $E$ into $V$ with the Tychonoff topology. If $cardV\le cardE$, then the set of all orthogonally additive surjections mapping $E$ into $V$ is dense in the space of all orthogonally additive functions from $E$ into $V$ with the Tychonoff topology.

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