Existing in Mathlib · Topology and analysis

Intermediate Value Theorem

If f is continuous from a to b and f(a) is below f(b), then no value between those endpoint outputs can be skipped: each one equals f(x) for some x between a and b.

intermediate values
continuity
closed intervals
connectedness
image inclusion

Exact theorem

Exact Mathlib statement

theorem intermediate_value_Icc {α : Type*} [ConditionallyCompleteLinearOrder α] [TopologicalSpace α] [OrderTopology α] [DenselyOrdered α] {δ : Type*} [LinearOrder δ] [TopologicalSpace δ] [OrderClosedTopology δ] {a b : α} (hab : a ≤ b) {f : α → δ} (hf : ContinuousOn f (Set.Icc a b)) : Set.Icc (f a) (f b) ⊆ f '' Set.Icc a b

The theorem at a glance

Intermediate Value Theorem — exact interval-image form

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

Editorial theorem poster states the ordered closed-interval hypotheses, shows a continuous curve meeting an intermediate horizontal level, and explains the four-step preimage conclusion. — The selected Mathlib declaration states the increasing endpoint-value orientation as an interval-image inclusion.

Statement structure

Statement and scope

Read the exact Mathlib declaration

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

Scientific figure illustrating: Continuity forces each value between f(a) and f(b) to occur at some point of the closed interval. — Continuity forces each value between f(a) and f(b) to occur at some point of the closed interval.

Why it matters

A mathematical landmark

The Intermediate Value Theorem captures the no-jumps consequence of continuity. It is central to real analysis, root existence, connectedness, and numerical bracketing arguments.

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 target indexes Mathlib's ordered interval-image inclusion for a continuous function on [a,b]: every value between f(a) and f(b) occurs when a is at most b. The selected orientation uses Icc(f a)(f b), so it does not separately state the reversed-endpoint form.

The selected declaration fixes the endpoint-value orientation from f(a) to f(b).
Reversed and unordered endpoint-value forms are separate Mathlib declarations.
ProofAtlas is indexing an existing Mathlib theorem, not claiming new mathematics.

Source and local evidence

Where the theorem comes from

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