A. Prove that a polynomial function y=P(x) is continuous at...

a. Prove that a polynomial function $y=P(x)$ is continuous at every number $x$. Follow these steps: i. Use Properties 2 and 3 of continuous functions to establish that the function $g(x)=x^{n}$, where $n$ is a positive integer, is continuous everywhere. ii. Use Properties 1 and 5 to show that $f(x)=c x^{n},$ where $c$ is a constant and $n$ is a positive integer, is continuous everywhere. iii. Use Property 4 to complete the proof of the result. b. Prove that a rational function $R(x)=p(x) / q(x)$ is continuous at every point $x$ where $q(x) \neq 0$.

a. Prove that a polynomial function $y=P(x)$ is continuous at every number $x$. Follow these steps:
i. Use Properties 2 and 3 of continuous functions to establish that the function $g(x)=x^{n}$, where $n$ is a positive integer, is continuous everywhere.
ii. Use Properties 1 and 5 to show that $f(x)=c x^{n},$ where $c$ is a constant and $n$ is a positive integer, is continuous everywhere.
iii. Use Property 4 to complete the proof of the result.
b. Prove that a rational function $R(x)=p(x) / q(x)$ is continuous at every point $x$ where $q(x) \neq 0$.

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Key Concepts

Continuity of Rational Functions

A rational function is defined as the quotient of two polynomials. Since both the numerator and denominator are continuous everywhere and the quotient of two continuous functions is continuous wherever the denominator is nonzero, a rational function is continuous at every point where its denominator does not vanish.

Properties of Continuous Functions

Certain operations preserve continuity: the sum, product, scalar multiplication, and composition of continuous functions remain continuous. These properties allow us to build complex continuous functions from simpler ones and are instrumental in proving the continuity of polynomial and rational functions.

Definition of Continuity

A function is continuous at a point if the limit of the function as the input approaches that point equals the function's value at that point. This foundational concept in calculus is used to determine whether functions behave 'smoothly' at and around specific points.

Continuity of Polynomial Functions

A polynomial function, being a finite sum of terms of the form c·x^n, is continuous because each term x^n is continuous for any positive integer n, and multiplying by a constant and adding such functions both preserve continuity. Thus, every polynomial is continuous everywhere.

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