Prove that a polynomial function y px is continuous at

Question

Question: 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) = xⁿ, where n is a positive integer, is continuous everywhere.

(ii) Use Properties 1 and 5 to show that f(x) = cxⁿ , 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) 0.