When a scalar field $f: S \subset \mathbb{R}^n \rightarrow \mathbb{R}$ is differentiable at a point $\textbf{a}$, and $f(x, y) = c$ is a level surface of this function, we can prove that $\nabla f(\textbf{a})$ is perpendicular at $\textbf{a}$ to every curve lying on this level surface passing through $\textbf{a}$. The tangent vectors at $\textbf{a}$ of this family of curves determine a plane with $\nabla f(\textbf{a})$ as a normal vector- and we call this plane a tangent plane.
This is what I've concluded is the definition of the tangent plane to a surface at a point $\textbf{a}$- if you consider the family of curves $\textbf{r}(t)$ lying on this surface passing through $\textbf{a}$, and the tangent vectors $\textbf{r}'(t)$ at $\textbf{a}$ lie on the same plane, this plane is termed the tangent plane.
Correct me if I'm wrong. I haven't found a definition of a tangent plane which didn't involve talking about differentiability.
Such a tangent plane will exist if $f$ is differentiable at $\textbf{a}$, but how do we show that such a plane will not exist if $f$ is not differentiable at $\textbf{a}$ (if we stick to the 'definition' of a tangent plane I've given above)?