On metric space valued functions of bounded essential variation

Let $\emptyset \ne T \subset \mathbb{R}$ and let $X$ be a metric space. For an ideal $\mathcal{J}\subset \mathcal{P}(T)$ and a function $f:T\to X$, we define the essential variation $V^{\mathcal{J}}_{ess}(f, T)$ as the in mum of all variations $V (g, T)$ where $g:T\to X, g = f$ on $T\setminus E$, and $E \in \mathcal{J}$. We show that if $X$ is complete then the essential variation of $f$ is equal to inf$\{V (f; T\setminus E) : E \in \mathcal{J}\}$. This extends former theorems of that type. We list some consequences that are analogues to the recent results by Chistyakov. Some examples of di erent kinds of essential variation are also investigated.