Order of continuity

As explained in Geometry for Programmers

The regularity class requiring continuous first derivatives in addition to continuity.

Order of continuity measures how many derivatives of a function remain continuous. A continuous function whose derivative is not continuous is C0; when its first derivative is continuous, it is C1; when its first and second derivatives are continuous, it is C2. More generally, Cn requires continuity through the nth derivative. section 5.2.1

For piecewise functions, continuity makes neighboring pieces meet at the same value. C1 smoothness additionally makes their derivatives agree, removing sharp corners. C2 smoothness also preserves continuous curvature, which is useful when joining engineered curves. section 5.2.1 section 5.2.4

Two piecewise joins

The pieces x² + 1 and 1 − x² meet at x = 0 with equal values and equal derivatives, producing a smooth join. By contrast, f(x) = x − 0.5 for x < 1 and f(x) = 1.5 − x for x ≥ 1 remains continuous but has a derivative mismatch and is not smooth. section 5.2.1

Figure 5.11 A piecewise function, smooth and continuous

Compare the smooth C1 piecewise graph with the continuous but cornered graph in b2 to see how derivative matching changes geometry. section 5.2.1

How Geometry for Programmers covers it

How deep it goes

Defines C0 through Cn and connects C1 to corners and C2 to curvature, but stops short of formal proofs.

Examples

Uses quadratic and linear piecewise functions, joined curves, and graphs illustrating C0, C1, curvature, and corner formation.

What it assumes

Assumes continuity and derivatives, while developing the hierarchy from C0 through Cn and relating derivatives to curve geometry.

What you can do after

Evaluate piecewise joins by checking whether neighboring values and successive derivatives coincide at their joint points.

How it connects

Connects continuity, derivatives, piecewise functions, parametric curves, smoothness, and curvature.

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Related concepts

• basisconfigured by · A basis is a linearly independent set of vectors whose linear combinations represent every vector in a vector space.
• smoothnessmeasures · Smoothness is the property of changing gradually, so nearby inputs produce similar outputs without discontinuities or abrupt changes.

This explainer was generated by Manning's knowledge graph from the text of Geometry for Programmers. The section links point to the parts of the book it draws on.