Confusion regarding the definition of uniform continuous functions on metric spaces.

Confusion regarding the definition of uniform continuous functions on
metric spaces.

What exactly is a uniform continuous function on a metric space?
My book says $f:X\to Y$ is uniform continuous if $\forall
\epsilon\in\Bbb{R}$, for any points $x,y\in X$, there exists a constant
$\delta$ such that $\rho(f(x),f(y))<\epsilon$ iff $d(x,y)<\delta$.
Is this equivalent to the condition that for every point $p\in X$, and
$\epsilon\in \Bbb{R}$, there exists a common $\delta$ such that
$|f(p)-\{f(|x-a|<\delta)\}|<\epsilon$?
Thanks in advance!