Symmetry of Second Derivatives

Definition

Consider a function $f(\mathbf{x}):\mathcal{D}\subset {\mathbb{R}}^n\to \mathbb{R}$ where $\mathbf{x}=x_1,x_2,\dots ,x_n$ If the all first and second partial derivatives of $f$ exists and continuous in a Neighborhood of the point $\mathbf{x}$, then $\frac{{\partial }^2}{\partial x_i\partial x_j}f(\mathbf{x})=\frac{{\partial }^2}{\partial x_j\partial x_i}f(\mathbf{x}),\forall i,j\in \{1,\dots ,n\}$