Hyers–Ulam stability

Remarks

Example: the Cauchy equation $f(x + y) = f(x) + f(y)$ for maps from a normed space to a Banach space is Hyers–Ulam stable with $K = 1$; here $k = 2$, $E_1(f)(x, y) = f(x + y)$ and $E_2(f)(x, y) = f(x) + f(y)$. Non-example: the periodicity equation $f(x + 1) = f(x)$ for maps $\R \to \R$ is not stable. For $\delta > 0$ the map $f(x) = \delta x$ satisfies $\abs{f(x + 1) - f(x)} = \delta$. If $g$ is $1$-periodic, then $f(x + n) - g(x + n) = f(x) - g(x) + n\delta$ for every integer $n$, so $f - g$ is unbounded and no $K$ works. Some authors use a weaker form: for every $\varepsilon > 0$ there is $\delta > 0$ such that a $\delta$-approximate solution lies within $\varepsilon$ of a solution. In the Hyers–Ulam–Rassias form, the constant bound $\delta$ is replaced by a control function such as $\theta(\norm{x}^p + \norm{y}^p)$.