  Given a continuum $X$, for each $A\subseteq X$, the Jones' set function
$\mathcal{T}$ is defined by $\mathcal{T}(A)=\{x\in X : \text{for each
subcontinuum }K\text{ such that }x\in \textrm{Int}(K), \text{ then }K\cap
A\neq\emptyset\}.$ We show that $\mathcal{D}=\{\mathcal{T}(\{x\}):x\in X\}$ is
decomposition of $X$ when $\mathcal{T}$ is continuous. We present a
characterization of the continuity of $\mathcal{T}$ and answer several open
questions posed by D. Bellamy.
