Rademacher's theorem

Every Lipschitz function on an open subset of Euclidean space is differentiable almost everywhere with respect to Lebesgue measure.

Lipschitz continuity: A function is Lipschitz continuous if one constant bounds the ratio of output distance to input distance for every pair of points. This regularity condition is the theorem’s central hypothesis.
Euclidean space: Euclidean space is a real coordinate space equipped with its usual distance and geometry. The theorem concerns Lipschitz functions defined on open subsets of this space.
Geometric measure theory: Geometric measure theory studies geometric objects using measure, integration, and generalized notions of surface area. Lipschitz parametrizations become differentiable almost everywhere, enabling tangent and area analysis.
Weierstrass function: The Weierstrass function is a continuous function that is nowhere differentiable. It shows continuity alone cannot replace the theorem’s Lipschitz condition.
Lebesgue differentiation theorem: A locally integrable function agrees almost everywhere with the limit of its local averages as the averaging regions shrink. Differentiation arguments use local averages to control behavior near almost every point.
Open set: A subset of a topological space is open when every point has a neighborhood contained in the subset. Openness gives each domain point room for local difference quotients in every direction.
Area formula: The area formula relates the integral of a Jacobian over a domain to the measure of its image under a Lipschitz map. Almost-everywhere differentiability supplies the derivative used in its Jacobian.
Cantor function: The Cantor function is a continuous nondecreasing function that is constant on each interval removed in constructing the Cantor set. Its derivative is zero almost everywhere despite the function’s nonconstant total change.
Lebesgue measure: Lebesgue measure assigns a consistent notion of volume to subsets of Euclidean space, including many irregular sets. It specifies what ‘almost everywhere’ means in the theorem.
Almost everywhere: A property holds almost everywhere when the points where it fails form a set of measure zero. This phrase states the precise extent of the theorem’s differentiability guarantee.

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