##  [Regular Value](/regular-value-0) 

 Definition

For a smooth map f:M→N between manifolds, a point y in N is a regular value if for every x in f^{-1}(y) the differential Df_x is surjective; equivalently all preimage points are regular points.

 

 

 

 

 

 





## Principle

Principle

By the regular value theorem (preimage/submersion theorem), the preimage f^{-1}(y) of a regular value is a smooth submanifold of M of codimension equal to dim N; regular values are generic by Sard's theorem.

 

 

 

 

 





## Demonstration

Demonstration

For f: R^2 → R given by f(x,y)=x^2+y^2-1, the value 0 is regular because the gradient ∇f is nonzero on the circle f^{-1}(0), so the circle is a 1-dimensional submanifold.

 

 

 

 

## Misapplication

Misapplication

Treating critical values as regular and therefore asserting manifold structure or wrong dimension counts for level sets without checking surjectivity of the derivative.

 

 

 

 

 





## Consequence

Consequence

When y is regular, f^{-1}(y) is a smooth embedded submanifold with predictable dimension and local charts; small perturbations of f preserve regularity of nearby values.

 

 

 

 

## Reversal

Reversal

A critical value has at least one preimage point where the derivative fails to be surjective; its preimage may have singularities or change topology under perturbation.

 

 

 

 

 





## Boundary

Boundary

Requires f to be differentiable of class at least C^1 between smooth manifolds; notion does not apply to non-differentiable maps or to arbitrary topological maps.

 

 

 

 

 





## Semantic Tension

Semantic Tension

Closely related to 'regular point' and to transversality; tension appears when distinguishing a regular value (property of the target point) from a regular point (property of a preimage point) and from transverse intersections.

 

 

 

 

 





## Synthesis

Synthesis

A regular value is a target value whose entire preimage consists of points where the derivative is surjective, guaranteeing that the preimage is a smooth submanifold of the expected codimension.