Solutions

a) This is the negation of the definition of continuity.

b) The negation of the definition of continuity is:

$$\exists \varepsilon>0,\ \forall \delta>0,\ \exists x \quad |x-a|<\delta \land |f(x)-f(a)|\geq \varepsilon.$$

Unpacking the second inequality, we have

$$\begin{align*} f(x)-f(a)\geq \varepsilon &\implies f(x)\geq \varepsilon+f(a),\\ \text{or}\qquad f(a)-f(x)\geq \varepsilon &\implies f(x)\leq -\varepsilon+f(a). \end{align*}$$