Negating logical statement of continuity I need help negating the following statement please, or if anyone could help putting it into words:
$$\forall \epsilon > 0 (\exists d>0(\forall x_0 (\forall x(|x - x_0| < d \implies |f(x) - f(x_0)| < \epsilon))))$$

The negation in words is: there exists a positive $\epsilon$ such that for all positive $\delta$ there exist $x_0$ and $x$ such that the distance between $x$ and $x_0$ is less than $\delta$, but the distance between $f(x)$ and $f(x_0)$ is at least $\epsilon$.
Symbolically, just push negation inside quantifiers by rewriting $\lnot\exists$ as $\forall\lnot$ and $\lnot\forall$ as $\exists\lnot$ and then use de Morgan's laws for the propositional connectives to rewrite, e.g., $\lnot(A \implies B)$ as $A \land \lnot B$.
