**Lemma** By negating this definition for [[Continuous Function (Epsilon-Delta Definition)|continuity]] we get that a function $f: E \to \mathbb{R}$ is discontinuous at $c \in E$ if there exists $\epsilon > 0$ such that for every $\delta> 0$ there exists an $x\in E$ such that $|x-c| < \delta \; \text{ and }\; |f(x) - f(c)| \geq \epsilon$ Note that we can either use this definition to prove that a function is discontinuous at a point or we can use this definition [[Discontinuous Function (Sequence Definition)]]. See [[Examples of Discontinuous Functions]].