Mathematics
Differentiability of Functions of Several Variables
Quick fact
A function of several variables can have all its partial derivatives at a point and still not be differentiable there. In fact, it can even have derivatives in every possible direction, yet fail to be differentiable—because differentiability requires a single linear approximation that works for all directions at once.
Why this is interesting
You've probably heard that a function is differentiable if you can find its partial derivatives. But in higher dimensions, that rule can completely mislead you—there are functions with partial derivatives at a point that aren't even continuous there. So what does differentiability really mean for a function of several variables?