E. Continuity of Polynomial and Rational Functions

Given a linear function f(x) = mx + b, we can add a square term, and get a quadratic function g(x) = ax² + f(x) = ax² + mx + b. We can continue adding terms of higher degrees, e.g. we can add a cube term and get h(x) = cx³ +g(x) = cx³ +ax² +mx+b, and so on. f(x), g(x), and h(x) are all special cases of a polynomial function.

Note that although an 6= 0, the remaining coefficients a _n−1, a _n−2, . . . , a₁, a₀ can very well be 0.

Rational Functions

Just as rational numbers are defined in terms of quotients of integers, rational functions are defined in terms of quotients of polynomials.