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.