As we know that $\epsilon-\delta$ definition of continuity between metric spaces $X$ and $Y$ can be stated as follows:
A map f:$(X, d_X)\rightarrow (Y, d_Y)$ is said to be continuous at a point $p\in$ X if for a given $\epsilon >0$, $\exists$ $\delta >0$ such that $d_X(p,x)< \delta\Rightarrow d_Y(f(p),f(x))< \epsilon $
I need help to understand this definition. Can we interpret it geometrically? How this definition is related with the definition of continuity of real variables?