The thesis addresses the problem of stability of additive transformations and isometries on euclidian spaces. Cauchy's functional equation is solved in the class of continious transformations. We define $\delta$-additive transformations for $\delta > 0$ and prove, that for every $\delta$-additive transformation there exists exactly one additive transformation, which approximates the former with norm of difference being less or equal to $\delta$. If we are dealing with $\delta$-additive transformation of $k$-th order for $k \ge 2$ the same property holds with a better approximation constant. We prove some properties of isometries and define $\delta$-isometries for $\delta > 0$. We prove that every $\delta$-isometry can be approximated by an isometry with the norm of difference being less than $10\delta$.
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