Alphabeta Math
CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-02
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Polynomials are uniformly dense in C([a,b],R) for every closed interval

Statement

For a≤b, every continuous real function on [a,b] is a uniform limit of polynomials.

Facts & Assumptions

Given: f∈C([a,b],R) and ε>0.

[L1]

Polynomials are uniformly dense on [0,1] (Polynomials are uniformly dense in C([0,1],R)).

[L2]

Closed intervals have the endpoint convention of Intervals of R: the nine order-convex forms, nondegeneracy, and length.

Proof

technique · direct
1.1

If a=b, the constant polynomial q(x)=f(a) agrees with f on the singleton interval.

givenL2algebra
1.2

Now suppose a<b and define g(t)=f(a+(b−a)t) on [0,1].

givenL2construct
1.3

Choose a polynomial p with ∣p(t)−g(t)∣<ε on [0,1] by [L1].

L1choose
2.1

Then q(x)=p((x−a)/(b−a)) is a polynomial and satisfies ∣q(x)−f(x)∣<ε on [a,b]. Together with step 1.1 this proves both cases.

step 1.1step 1.2step 1.3algebra∎

Depends on

Used by

Dependency tree · two levels

7 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