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Every continuous function defined on an open interval can be uniformly approximated by polynomials.

a. True
b. False

User Bardicer
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1 Answer

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Final answer:

True. Every continuous function defined on an open interval can be uniformly approximated by polynomials.

Step-by-step explanation:

True. Every continuous function defined on an open interval can be uniformly approximated by polynomials. This is known as the Stone-Weierstrass theorem in analysis.

The Stone-Weierstrass theorem states that for any continuous function f(x) defined on a closed interval [a, b], and any ε > 0, there exists a polynomial P(x) such that |f(x) - P(x)| < ε for all x in [a, b]. In other words, the polynomial P(x) can approximate the function f(x) uniformly on the interval [a, b].

Polynomial approximations are important in many areas of mathematics and engineering, as they provide a way to represent and work with continuous functions numerically. They have applications in physics, computer science, statistics, and many other disciplines.

User Naxi
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