Let's consider the function $f:\mathbb R^n\rightarrow\mathbb R$. According to Wikipedia, the gradient of $f$ is defined as the unique vector field whose dot product with a unit vector $\mathbf v$ is the directional derivative of $f$ in the direction of $\mathbf v$: $$D_\mathbf v f = \nabla f\cdot\mathbf v.$$ I've been wondering how do we know that such a vector field exists and is unique.
Any help is appreciated, thanks.