Let $f:\mathbb{R}^n\to \mathbb{R}$ be Lipsticz function. The directional derivative in the direction of $\mathbf{v}$ is given by $$ D_\mathbf{v}f(\mathbf{x})=\lim_{t\to 0}\dfrac{f(\mathbf{x}+t\mathbf{v})-f(\mathbf{x})}{t}. $$ Define a set $$ D_\mathbf{v}=\{ \mathbf{x}\in \mathbb{R}^n:D_\mathbf{v}f(\mathbf{x})\text{ exists } \} .$$ Show that $D_\mathbf{v}$ is measurable.
Please help me in proving that.