# Weierstrass Approximation Theorem

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## Theorem

Let $f$ be a real function which is continuous on the closed interval $\Bbb I$.

Then $f$ can be uniformly approximated on $\Bbb I$ by a polynomial function to any given degree of accuracy.

## Proof

## Also known as

This result is also seen referred to as **Weierstrass's theorem**, but as there are a number of results bearing Karl Weierstrass's name, it makes sense to be more specific.

## Source of Name

This entry was named for Karl Weierstrass.

## Historical Note

The Weierstrass Approximation Theorem has been demonstrated to have far-reaching and important effects in every aspect of the field of analysis.

It has also been given a significant generalisation by Marshall Harvey Stone.

## Sources

- 1992: George F. Simmons:
*Calculus Gems*... (previous) ... (next): Chapter $\text {A}.33$: Weierstrass ($1815$ – $1897$) - 1989: Ephraim J. Borowski and Jonathan M. Borwein:
*Dictionary of Mathematics*... (previous) ... (next): Entry:**Weierstrass approximation theorem** - 1998: David Nelson:
*The Penguin Dictionary of Mathematics*(2nd ed.) ... (previous) ... (next): Entry:**Weierstrass's theorem** - 2008: David Nelson:
*The Penguin Dictionary of Mathematics*(4th ed.) ... (previous) ... (next): Entry:**Weierstrass's theorem**