LemmaStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (z-ai/glm-5.2)audited 2026-07-31

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A finite concatenation of straight segments in $R^{n}$ is a continuous path

Statement

Let $v_{0}, \dots, v_{m} \in R^{n}$ and $0 = t_{0} < \dots < t_{m} = 1$ . The affine pieces joining $v_{i - 1}$ to $v_{i}$ define a continuous map $[0, 1] \to R^{n}$ . If every piece lies in a subset $A$ , the map is a polygonal path (Polygonal paths and polygonally connected subsets of $R^{n}$ ) in $A$ from $v_{0}$ to $v_{m}$ .

Facts & Assumptions

Given: Vertices $v_{0}, \dots, v_{m} \in R^{n}$ and a partition $0 = t_{0} < \dots < t_{m} = 1$ .

[L1]

A finite family of closed sets covering a space pastes continuous restrictions to a continuous map (Continuity may be checked on any open cover, and on any finite closed cover; composites of continuous maps are continuous, claim 3).

[L2]

A map whose values lie in a subspace is continuous into that subspace exactly when its composite with the inclusion into the ambient space is continuous (Subspace topology: the traces of the open sets, its closed sets and its bases, the continuity of the inclusion, and the characteristic property of a map into a subspace).

[L3]

A map into $R^{n}$ is continuous if and only if all of its coordinate functions are continuous (A vector-valued function has a limit, or is continuous, if and only if each of its components does; with the algebra of continuous vector-valued functions, clause 1).

Proof

technique · constructive

1.1

On $[t_{i - 1}, t_{i}]$ define $γ_{i} (t) : = ((t_{i} - t) / (t_{i} - t_{i - 1})) v_{i - 1} + ((t - t_{i - 1}) / (t_{i} - t_{i - 1})) v_{i}$ . Each coordinate is an affine real function of $t$ , hence continuous.

L3construct

2.1

At every shared endpoint $t_{i}$ , the adjacent formulas both give $v_{i}$ , so the pieces define one function $γ : [0, 1] \to R^{n}$ .

step 1.1construct

3.1

Each interval $[t_{i - 1}, t_{i}]$ is closed in $[0, 1]$ , the finitely many intervals cover it, and each restriction of $γ$ is continuous by step 1.1. Thus $γ$ is continuous by [L1].

L1step 1.1step 2.1

4.1

If the pieces lie in $A$ , then $γ$ takes values in $A$ . Its composite with the inclusion $A ↪ R^{n}$ is the continuous map of step 3.1, so [L2] makes it continuous into the subspace $A$ . Its endpoints are $v_{0}, v_{m}$ , hence it is a path in $A$ .

step 3.1L2discharge-construct∎

Depends on

Used by

Dependency tree · two levels

35 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Path-connected space (standard reference, not scraped)
Pasting lemma (Wikipedia) (standard reference, not scraped)
Polygonal chain (standard reference, not scraped)

A finite concatenation of straight segments in RnRn is a continuous path