differentiability (multivariable)

The stronger condition, beyond both partials existing, that a linear plane approximates $f$ well from EVERY direction at a point.

$f$ is differentiable at $(a,b)$ if the linear function $L(x,y) = f(a,b) + f_x(a,b)(x-a) + f_y(a,b)(y-b)$ approximates $f$ well enough that the error vanishes FASTER than the distance to $(a,b)$ : $\lim_{(x,y)\to(a,b)} \frac{f(x,y) - L(x,y)}{\sqrt{(x-a)^2+(y-b)^2}} = 0$ . This is strictly stronger than " $f_x(a,b)$ and $f_y(a,b)$ both exist" — that weaker condition only requires good behavior along the two axis-aligned slices through $(a,b)$ , while differentiability (also called local linearity) requires $L$ to track $f$ from every direction at once. A sufficient condition: if $f_x$ and $f_y$ both exist and are continuous on an open region, $f$ is automatically differentiable there.

Examples

$f(x,y)=x^2-y^2$ is differentiable everywhere since $f_x=2x$ and $f_y=-2y$ are continuous polynomials.
$f(x,y)=\sqrt{x^2+y^2}$ has partials that exist away from the origin but the surface has a sharp cone point at $(0,0)$ that no axis-aligned slice reveals.