Increment Theorem

Definition

If the partial derivatives of a function are continuous, then $f$ is differentiable. $f(\mathbf{x}+\Delta \mathbf{x})-f(\mathbf{x})=\nabla f(\mathbf{x})\Delta \mathbf{x}+\mathbf{ϵ}\Delta \mathbf{x}$ where $\underset{\Delta \mathbf{x}\to 0}{\lim }ϵ=0$