Continuous Map Between Metric Spaces

Let $(X,d_X)$ and $(Y,d_Y)$ be metric spaces, let $A\subseteq X$, let $f:A\to Y$, and let $x\in A$. Let $\mathbb{R}$ be the set of real numbers with the order $\le$ of its ordered field structure, and for $a,b\in\mathbb{R}$ write $a<b$ to mean that $a\le b$ and $a\ne b$.

We say that $f$ is continuous at $x$ relative to $A$ if for every $\varepsilon\in\mathbb{R}$ with $0<\varepsilon$ there exists $\delta\in\mathbb{R}$ with $0<\delta$ such that every $y\in A$ satisfying $d_X(x,y)<\delta$ satisfies

We say that $f$ is continuous on $A$ if $f$ is continuous at $x$ relative to $A$ for every $x\in A$.