Let $\displaystyle f: \mathbb{R}^n \to \mathbb{R}$, and suppose that some pair of f's mixed partials don't commute at some point p:

$\displaystyle \frac{\partial^2 f}{\partial x_i\partial x_j}(p) \neq \frac{\partial^2 f}{\partial x_j \partial x_i}(p).$

Then one of f's second derivatives is discontinuous at p. Is there any way to recover which one is discontinuous, and from which direction? That is, to find a k, l, and f(t) (with f(0) = p) such that $\displaystyle g(t) = \frac{\partial^2 f}{\partial x_k \partial x_l}[f(t)]$ is discontinuous at t=0?