The definition for uniform continuity for a function $f : X \to Y$ is that
for all $\epsilon > 0$, there exists a $\delta > 0$ such that $d_Y(f(p),f(q)) < \epsilon$ for all $p,q \in X$ such that $d_X(p,q) < \delta$.
Mathematically, we can write this definition as
$\forall \epsilon > 0, \exists \delta > 0 \textrm{ such that } d_Y(f(p),f(q) < \epsilon\ \forall p,q \in X \textrm{ for which } d_X(p,q) < \delta.$
I have learned that when negating a statement, one switches the quantifiers, and reverses any equality/inequality statements in the conclusion.
Following these rules, the definition for not uniformly continuous would be
$\exists \epsilon > 0\ \forall \delta > 0 \textrm{ such that } d_Y(f(p),f(q)) \ge \epsilon\ \exists p,q \in X \textrm{ for which } d_X(p,q) < \delta.$
This, however makes little sense as written. I believe that I would be better served writing this as
$\exists p,q \in X \textrm{ such that } \forall \delta > 0 \textrm { for which } d_X(p,q) < \delta, \exists \epsilon > 0 \textrm{ such that } d_Y(f(p),f(q)) \ge \epsilon.$
My question is: does my latter define criteria for which a function is not uniformly continuous? In other words, is this a proper negation of the definition? I believe that it should provide a criteria for a function not being uniformly continuous, but is this the same thing as the wholesale negation of the definition?