Hyers–Ulam stability

Summary

A functional equation is Hyers–Ulam stable if each approximate solution is near an exact one. Precisely, a function that satisfies it up to an error $\delta$ lies within $K\delta$ of a solution, for a fixed constant $K$. The notion comes from a question of Ulam on approximate homomorphisms. Hyers' theorem shows that the Cauchy equation $f(x+y) = f(x) + f(y)$ is stable, with $K = 1$, for maps from a normed space to a Banach space. Stability results justify replacing an approximate solution by an exact one.