Sunday, February 01, 2009

Polynomials are continuous

For a definition of polynomials, see Definition 1, here. For a definition of continuous functions, see Definition 1, here.

Lemma 1: f(x)=x is continuous


(1) Let ε be any arbitrary value.

(2) Let δ = ε

(3) For any point c, it is clear that if x lies in (c - δ, c + δ), then f(x)=x lies in (f(c) - ε, f(c) + ε )


Corollary 1.1 : Polynomials are continuous


(1) The function f(x)=x is continuous. [See Lemma 1 above]

(2) Since the product of continuous functions is continuous [See Lemma 3, here], then f(x)=xn where n is a positive integer is also continuous.

(3) Since f(x)=C is continuous [See Lemma 1, here], it follows that any function of the form cxn is also continuous.

(4) Since the addition of continuous functions is continuous [See Lemma 2, here], it follows that any polynomial function is continuous since it consists of the form:

f(x) = c0 + c1x + c2x2 + ... + cnxn

where each ci is a constant.


Lemma 2: The Derivative of a polynomial is itself a polynomial


(1) The derivative of each term of a polynomial is itself a term of a polynomial [See the Lemma 2, here]

(2) So, it follows that the derivative itself is also a polynomial. [See Definition 1, here]


Corollary 2.1: The derivative of a polynomial is a continuous function.


This follows directly from Lemma 2 above and Corollary 1.1 above.



Nia said...

Dear Larry,

Great posts!!!

I am a UCSC senior Math student and I tutor intro to analysis.

I have found your post most through and useful for my students (and for myself as well for memory refreshment).

Please never ever remove them. It would be a great loss to humanity if we couldn't access them anymore.

Thanks for putting your time and energy into this blog!!

Nia said...

oops I missed an "o" from thorough.