# Regular value

## Definition

### Symbol-free definition

Given a smooth map of differential manifolds, a point in the image manifold is termed a regular value for the smooth map if every point in its inverse image is a regular point, i.e. if the map from the tangent space at any point in the inverse image, is surjective.

Note that any point whose inverse image is empty, is by definition a regular value.

### Definition with symbols

Let $f:M \to N$ be a smooth map of differential manifolds. A point $p \in N$ is termed a regular value of $f$ if for every $m \in f^{-1}(\{p\})$, the induced map $(df)_m: T_mM \to T_pN$, is surjective.