Existing in Mathlib · Analysis

Taylor's Theorem with Lagrange Remainder

For x₀ < x, if f has the selected finite-order continuity and differentiability properties on the closed interval and its interior, then the error of its degree-n Taylor polynomial at x equals the standard Lagrange remainder evaluated at some interior point x'.

Taylor polynomials
Lagrange remainder
iterated derivatives
differentiability on intervals
mean value theorem
factorial scaling

Exact theorem

Exact Mathlib statement

theorem taylor_mean_remainder_lagrange {f : ℝ → ℝ} {x x₀ : ℝ} {n : ℕ} (hx : x₀ < x) (hf : ContDiffOn ℝ n f (Icc x₀ x)) (hf' : DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)) : ∃ x' ∈ Ioo x₀ x, f x - taylorWithinEval f n (Icc x₀ x) x₀ x = iteratedDerivWithin (n + 1) f (Icc x₀ x) x' * (x - x₀) ^ (n + 1) / (n + 1)!

The theorem at a glance

Taylor's theorem with Lagrange remainder at a glance

A visual guide to the theorem's hypotheses, structure, and conclusion; the exact statement gives the formal detail.

An ivory calculus poster places a smooth function and finite Taylor polynomial above the ordered closed interval from x₀ to x, with a highlighted x' strictly inside, the remainder gap at x, an interior derivative medallion, and the exact hypotheses and Lagrange remainder identity. — Under the selected closed-interval continuity and open-interval differentiability hypotheses, some interior point realizes the exact Lagrange remainder of the finite Taylor polynomial.

Accessible poster text

Read the headline and panel text

ANALYSIS · DIFFERENTIAL CALCULUS

TAYLOR'S THEOREM

WITH LAGRANGE REMAINDER

f : ℝ → ℝ · n : ℕ · x₀ < x

ContDiffOn ℝ n f (Icc x₀ x)

DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x)

THERE EXISTS x' ∈ Ioo x₀ x SUCH THAT

f x − taylorWithinEval f n (Icc x₀ x) x₀ x

= iteratedDerivWithin (n + 1) f (Icc x₀ x) x'

× (x − x₀)^(n + 1) / (n + 1)!

THE FINITE TAYLOR REMAINDER IS REALIZED AT AN INTERIOR POINT

HOW THE CHECKED ROUTE MOVES

1 · Set g(t) = (x − t)^(n + 1)

2 · Its derivative is nonzero for t ∈ Ioo x₀ x

3 · Apply the general mean-remainder theorem

4 · Simplify to the Lagrange remainder formula

EXACT SCOPE

Finite Taylor polynomial with within-derivatives. No analytic-series claim. The witness need not be unique.

The detailed visual description covers the diagram and its local mathematical labels.

Statement structure

Statement and scope

Read the exact Mathlib declaration

This map summarizes the statement's structure; the exact Mathlib declaration remains authoritative.

Two curves share their left contact, separate at the right endpoint, and leave a vertical remainder gap; below, one emerald point strictly inside an open interval anchors a finite derivative fan and a gold guide to the remainder. — A finite Taylor jet leaves a remainder at the right endpoint that the theorem realizes through a derivative at some strictly interior witness.

Why it matters

A mathematical landmark

Taylor's theorem is a central bridge between local derivative data and finite polynomial approximation. Mathlib's selected declaration is especially instructive because its exact interval hypotheses and within-derivative formulation expose the finite mean-value mechanism while keeping the interior witness and the boundary of the result explicit.

ProofAtlas record

What has been checked

Upstream indexedPinned source bytes verified locally

Locally reproducedExact upstream declaration replayed

Reviewed pageCurrent public presentation reviewed

Accepted Atlas resultNot recorded for the preferred artifact

Mathlib is the source of the theorem; the local Lean replay and page review are separate.

Claim boundary

No new theorem is claimed

This page indexes Mathlib's selected Lagrange-remainder declaration for a real-valued function on ordered endpoints x₀ < x. Its exact assumptions are ContDiffOn ℝ n f (Icc x₀ x) together with DifferentiableOn ℝ (iteratedDerivWithin n f (Icc x₀ x)) (Ioo x₀ x). It concludes that some x' strictly inside Ioo x₀ x realizes the exact finite Taylor remainder through iteratedDerivWithin (n + 1) on the same closed interval. It does not use the nearby corollary's single ContDiffOn ℝ (n + 1) hypothesis, assert uniqueness of x', handle reversed or equal endpoints, or state an analytic-series or convergence theorem.

The selected declaration assumes ContDiffOn ℝ n f on Icc x₀ x and differentiability of the nth iterated within-derivative on Ioo x₀ x; it does not replace them by the nearby corollary's single ContDiffOn ℝ (n + 1) assumption.
The theorem concerns the finite polynomial taylorWithinEval and an exact remainder identity; it does not assert analyticity, an infinite Taylor-series identity, or convergence as n tends to infinity.
The conclusion gives at least one x' strictly inside Ioo x₀ x; it does not make that witness unique, canonical, or equal to an endpoint.
The selected orientation is x₀ < x and does not state the corresponding result for equal or reversed endpoints.
ProofAtlas is indexing an existing Mathlib theorem, and generated explanations and visuals are not proof evidence.

Source and local evidence

Where the theorem comes from

Existing declaration: taylor_mean_remainder_lagrange in mathlib
Relationship: The checked artifact is the upstream declaration itself
Local Lean evidence: artifact.library.mathlib.taylor-theorem.v001
Source: Open the pinned upstream reference