Lebesgue differentiation and absolute continuity

Let f be locally integrable on the real line. Almost every x is a Lebesgue point: the mean of |f(t)−f(x)| over (x−r,x+r) tends to zero as r decreases to zero. Consequently the unsigned-error control forces the signed average of f to approach f(x). Local integrability is enough because each bounded interval has finite integral of |f|. The exceptional set may be nonempty, and modifying a representative on a null set can deliberately spoil the conclusion at selected points without changing its integrable equivalence class.

A step function illustrates an allowed exception. At a jump from zero to one, symmetric averages tend to one half, which need not equal the assigned value at the jump. Away from the jump, sufficiently small intervals lie on one side and averaging works immediately. The general theorem is much deeper: approximation by continuous functions is combined with a maximal-function estimate controlling where local averaging errors are large. A graph sampled at finitely many points cannot establish that almost-everywhere estimate for an arbitrary integrable function.

A function F on [a,b] is absolutely continuous if for every epsilon>0 there is a delta>0 such that any finite family of pairwise disjoint subintervals with total length below delta has total absolute F-increment below epsilon. This is stronger than ordinary uniform continuity, which controls one small interval at a time without summing many increments. Lipschitz functions are absolutely continuous, but an absolutely continuous function can have an unbounded derivative. The condition restricts accumulated variation rather than imposing one global slope bound.

If g is integrable on [a,b], the function F(x)=F(a)+∫aˣ g(t)dt is absolutely continuous. To see why, bound the sum of absolute increments over disjoint intervals by the integral of |g| over their union. Integrability makes that integral small when the union has sufficiently small measure: truncate |g| at a large cutoff and control its remaining integral tail. Lebesgue differentiation then gives F′=g almost everywhere. The construction supplies both the continuity property and the correct derivative statement.

The converse direction is not merely that F has an integrable derivative almost everywhere. The Cantor function is continuous and increasing from zero to one, yet its derivative is zero almost everywhere. Integrating that derivative returns zero and misses the function's total increase. Its variation is concentrated on the null Cantor set, so it is not absolutely continuous. This example separates ordinary continuity, bounded variation, almost-everywhere differentiability, and absolute continuity. The singular component is a measure-theoretic phenomenon, not a numerical differentiation error.

For an absolutely continuous function, null-set changes to its derivative do not affect the reconstruction integral. Null-set changes to the function itself are different: changing a single value can destroy continuity and therefore destroy absolute continuity, even though the function represents the same L1 class. Thus the fundamental theorem concerns a particular continuous representative reconstructed at every point, while its derivative is specified only almost everywhere. Keeping those roles separate is important in weak derivatives, differential equations, and one-dimensional Sobolev spaces.