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Conservative Electric Field

If the electric potential is continuous, then the electric field is conservative, meaning the line integral of E along a closed path is zero.
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The statement of the theorem

Let Ω⊂R3\Omega \subset \mathbb{R}^3 be a simply connected domain. A vector field E⃗:Ω→R3\vec{E}: \Omega \to \mathbb{R}^3 is defined as a conservative electric field if and only if there exists a scalar potential function ϕ:Ω→R\phi: \Omega \to \mathbb{R} (the electric potential) such that E⃗=−∇ϕ\vec{E} = -\nabla\phi. Equivalently, this condition is characterized by the vanishing of the curl of E⃗\vec{E}: \n\n∇×E⃗=0⃗in Ω\nabla \times \vec{E} = \vec{0} \quad \text{in } \Omega \n\nFurthermore, if E⃗\vec{E} is sufficiently smooth (i.e., E⃗∈C1(Ω)\vec{E} \in C^1(\Omega)), the potential ϕ\phi can be found by integrating the line integral along any path CC from a reference point AA to a point BB: \n\nϕ(B)−ϕ(A)=−∫ABE⃗⋅dl⃗\phi(B) - \phi(A) = -\int_{A}^{B} \vec{E} \cdot d\vec{l} \n\nThis implies that the line integral is path-independent, which is the fundamental physical definition of a conservative field.
Source: Wikipedia