Q:

What does this notation, $x>x_0(\epsilon)$, mean?

What does this notation, $x>x_0(\epsilon)$, mean?
I have seen this in several proofs and haven't quite figured it out.

A:

$x_0(\varepsilon)$ means that $x_0$ depend on $\varepsilon$. For example 
$$\forall\varepsilon>0, \exists x_0(\varepsilon)>0, \forall x>x_0(\varepsilon), \frac 1{1+x^2}\leq \varepsilon.$$
It's in particular useful when the $x_0$ may depend on several variable; in this case it shows on which variable the $x_0$ depends. An example is the difference between pointwise convergence of a sequence of functions $\{f_n\}$ defined on a set $S$:
$$\tag{PC}\forall \varepsilon,\forall x\in S,\exists n_0(\varepsilon,x),\, \forall n\geq n_0(\varepsilon,x),\quad |f_n(x)-f(x)|\leq\varepsilon,$$
and the uniform convergence on $S$:
$$\tag{UC}\forall \varepsilon,,\exists n_0(\varepsilon),\, \forall x\in S,\forall n\geq n_0(\varepsilon),\quad |f_n(x)-f(x)|\leq\varepsilon.$$
In these two case, this can be written without marking the dependence, since it's implicit. 

