Let $f:X \rightarrow Y$ (and let $A \subset X$ where $f$ is continuous.$)^{(1)}$
If $f$ is continuous on $A \equiv $ { $x\in X$: $\forall ε>0 , \exists δ>0$ such as that $d_Y(f(z),f(y))<ε ,\forall \ z,y \in B(x,δ)$}$^{(2)}$ (where $d_Y$ metric on Y)
My thoughts are that, the deffinition describes a uniformly continuous function, as for my doubts, they are created by the fact that the handscript I was reading claims that this definition is equal to continuous function. Also I am aware of the fact that a uniformly continuous function is a continuous one and this suggests that $A \subseteq (2)$ whereas the sentence above claims that $A \equiv (2) $