Differentiability of a Bivariate Function

Differentiability of bivariate functions is defined differently to standard, unary functions.

Definition §

Let $f(x,y)$ be a bivariate function and let $P=(x_0,y_0)$ be a point. $f$ is differentiable at $P$ if, and only if all the tangent lines of a surface at point $P$ form a Tangent Plane.

In simpler terms:

• A curve has one, and only one tangent line at a given point.
• A bivariate function has infinite curves at a point, and hence infinite tangent lines.
• ALL these tangent lines must have the same gradient, to form a tangent plane.

Since a tangent line is defined using the derivate, a tangent plane can only exist if both Partial Derivative exist. Thus, just like in unary functions, differentiability only exists if a function is continuous near the point.