differentiability in R^n

What should it mean for a multivariable function to be smooth at a point? The honest answer copies the one-variable idea but states it carefully: the function should hug a flat plane so closely that, as you zoom in, the gap between the function and that plane shrinks faster than the distance to the point. Differentiability is exactly the guarantee that such a perfectly fitting linear approximation exists.

Precisely, f: R^n -> R^m is differentiable at a if there is a linear map L (its total derivative) with f(a + h) = f(a) + L h + r(h) and lim as h -> 0 of |r(h)| / |h| = 0. The crucial demand is that the error vanishes relative to |h|, not merely that it vanishes. This single condition controls all directions of approach at once.

Two honest cautions. First, differentiable implies continuous, but not conversely. Second — and this surprises newcomers — the mere existence of all partial derivatives is far weaker than differentiability and does not even force continuity. A clean sufficient condition is that the partial derivatives exist near a and are continuous at a; then f is differentiable at a.