The Weierstrass Theorem is important in real analysis since it establishes that every continuous function defined on a closed interval [a, b] may be uniformly approximated by a polynomial function. In other words, a continuous function f on [a, b], being given any positive number ε and x in [a, b], there would always exist a polynomial function p such that the absolute difference between f(x) and p(x) is less than ε and x in

The Weierstrass Theorem is important because it has a number of applications in real analysis, including:

Proofs of the Weierstrass theorem in function spaces imply a much wider result on the approximation by such classes of functions as trigonometric and exponential ones.
It may be used to establish the existence of solutions to various differential equations.
It can be applied in designing numerical methods for approximately evaluating the values of continuous functions.
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