Differentiable

A function $f:{\mathbb{R}}^n\to {\mathbb{R}}^m$ is differentiable at point $a$ if its Linearization is a good approximation of $f$ near $a$

$$\underset{\vec{v}\to \vec{a}}{\lim }\frac{f(\vec{v})-L(\vec{v})}{\Vert \vec{v}-\vec{a}\Vert }=0$$

In particular if f has a unique tangent at the point; in 1 variable: a tangent line. In 2 variables a tangent plane.