Polynomial Functions

Polynomial functions are the most basic type of continuous functions and play an important role in calculus.

Understanding the continuity properties of polynomial functions is crucial for learning more complex function types.

Basic Properties

Polynomial functions have the following basic properties:

Domain: $\mathbb{R}$ (all real numbers)
Continuity: Continuous everywhere within the domain
Graph characteristics: Smooth curves, no jumps or breaks
Differentiability: Differentiable everywhere within the domain

Definition of Polynomial Functions

A function of the form $f(x) = a_n x^n + a_{n - 1} x^{n - 1} + \cdots + a_1 x + a_0$ is called a polynomial function, where $a_n, a_{n - 1}, \ldots, a_0$ are constants, and $n$ is a non-negative integer.

Common Polynomial Functions

1. Linear Functions

Form: $f(x) = ax + b$ ( $a \neq 0$ )

Example: $f(x) = 2x + 1$

Properties:

The graph is a straight line
Slope is $a$
Continuous everywhere on $\mathbb{R}$

2. Quadratic Functions

Form: $f(x) = ax^2 + bx + c$ ( $a \neq 0$ )

Example: $f(x) = x^2 + 2x + 1$

Properties:

The graph is a parabola
Opens upward when $a > 0$ , opens downward when $a < 0$
Continuous everywhere on $\mathbb{R}$

3. Cubic Functions

Form: $f(x) = ax^3 + bx^2 + cx + d$ ( $a \neq 0$ )

Example: $f(x) = x^3 - 3x + 2$

Properties:

The graph is a cubic curve
May have local extrema
Continuous everywhere on $\mathbb{R}$

Continuity Proof of Polynomial Functions

Theoretical Foundation

The continuity of polynomial functions is based on the following facts:

Constant functions are continuous: $f(x) = c$ is continuous on $\mathbb{R}$
Power functions are continuous: $f(x) = x^n$ is continuous on $\mathbb{R}$
Sum, difference, and product of continuous functions are continuous

Proof Approach

Let $f(x) = a_n x^n + a_{n-1} x^{n-1} + \cdots + a_1 x + a_0$

Each power function $x^k$ is continuous on $\mathbb{R}$
The constant multiple $a_k x^k$ is continuous on $\mathbb{R}$
The sum of finitely many continuous functions is continuous on $\mathbb{R}$
Therefore, polynomial functions are continuous on $\mathbb{R}$

Graph Characteristics of Polynomial Functions

Visualization Example

Exercises

Exercise 1

Determine the continuity of the function $f(x) = x^3 + 2x^2 - x + 1$ at $x = 0$ .

Reference Answer

Solution Approach: Polynomial functions are continuous everywhere within their domain.

Detailed Steps:

$f(x) = x^3 + 2x^2 - x + 1$ is a polynomial function
The domain of polynomial functions is $\mathbb{R}$
Polynomial functions are continuous everywhere on $\mathbb{R}$
Therefore, $f(x)$ is continuous at $x = 0$

Answer: The function is continuous at $x = 0$ .

Exercise 2

Determine the continuity of the function $f(x) = 3x^2 - 2x + 5$ at $x = 1$ .

Reference Answer

Solution Approach: Polynomial functions are continuous everywhere within their domain.

Detailed Steps:

$f(x) = 3x^2 - 2x + 5$ is a quadratic polynomial function
Polynomial functions are continuous everywhere on $\mathbb{R}$
Therefore, $f(x)$ is continuous at $x = 1$

Answer: The function is continuous at $x = 1$ .

Exercise 3

Given $f(x) = x^4 - 2x^3 + x^2 - 1$ , find the continuity interval of $f(x)$ .

Reference Answer

Solution Approach: The continuity interval of a polynomial function is its domain.

Detailed Steps:

$f(x) = x^4 - 2x^3 + x^2 - 1$ is a quartic polynomial function
The domain of polynomial functions is $\mathbb{R}$
Polynomial functions are continuous everywhere on $\mathbb{R}$
Therefore, the continuity interval is $\mathbb{R}$

Answer: Continuity interval is $\mathbb{R}$ .

Exercise 4

Given $f(x) = x^3 + ax^2 + bx + c$ , where $a, b, c$ are constants. If $f(x)$ is continuous at $x = 1$ , find the value of $a + b + c$ .

Reference Answer

Solution Approach: Polynomial functions are continuous everywhere, so the condition that $f(x)$ is continuous at $x = 1$ imposes no restrictions on $a, b, c$ .

Detailed Steps:

Polynomial functions are continuous everywhere on $\mathbb{R}$
Therefore, the continuity of $f(x)$ at $x = 1$ imposes no restrictions on $a, b, c$
$a + b + c$ can be any real number

Answer: $a + b + c$ can be any real number.

Exercise 5

Determine the continuity of the function $f(x) = x^5 - 3x^3 + 2x$ on $\mathbb{R}$ .

Reference Answer

Solution Approach: Polynomial functions are continuous everywhere within their domain.

Detailed Steps:

$f(x) = x^5 - 3x^3 + 2x$ is a quintic polynomial function
The domain of polynomial functions is $\mathbb{R}$
Polynomial functions are continuous everywhere on $\mathbb{R}$
Therefore, $f(x)$ is continuous on $\mathbb{R}$

Answer: The function is continuous on $\mathbb{R}$ .

Exercise 6

Adapted from 2025 Graduate Entrance Exam Mathematics I Question 1

Given the function $f(x) = x^3 + 2x^2 + ax + b$ , where $a, b$ are constants. If $f(x)$ is continuous at $x = 0$ , find the value of $b$ .

Reference Answer

Solution Approach: Polynomial functions are continuous everywhere in their domain. The continuity of $f(x)$ at $x=0$ is equivalent to $f(0)$ existing and equaling the limit.

Detailed Steps:

$f(0) = b$
Polynomial functions are continuous everywhere on $\mathbb{R}$ , so $f(0)$ necessarily exists
Therefore, $b$ can be any real number

Answer: $b$ can be any real number.

Exercise 7

Adapted from 2023 Graduate Entrance Exam Mathematics I Fill-in-the-Blank

Determine the continuity of the function $f(x) = x^4 - 4x^2 + 3$ at $x=1$ and $x=-1$ , and explain the reason.

Reference Answer

Solution Approach: Polynomial functions are continuous everywhere within their domain. Just substitute directly.

Detailed Steps:

$f(x)$ is a quartic polynomial function with domain $\mathbb{R}$
Polynomial functions are continuous everywhere on $\mathbb{R}$
Both $x=1$ and $x=-1$ belong to the domain

Answer: $f(x)$ is continuous at both $x=1$ and $x=-1$ .

Summary

Symbols Appearing in This Article

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\mathbb{R}$	Mathematical Symbol	Blackboard bold R (Real numbers)	Represents the set of real numbers, domain of polynomial functions

Chinese-English Glossary

Chinese Term	English Term	Phonetic	Explanation
多项式函数	polynomial function	/pɒlɪˈnəʊmiəl ˈfʌŋkʃən/	Function of the form $f(x) = a_n x^n + \cdots + a_0$
一次函数	linear function	/ˈlɪniə ˈfʌŋkʃən/	Function of the form $f(x) = ax + b$
二次函数	quadratic function	/kwɒˈdrætɪk ˈfʌŋkʃən/	Function of the form $f(x) = ax^2 + bx + c$
三次函数	cubic function	/ˈkjuːbɪk ˈfʌŋkʃən/	Function of the form $f(x) = ax^3 + bx^2 + cx + d$
实数集	real numbers	/riːl ˈnʌmbəz/	The set of all real numbers, denoted as $\mathbb{R}$
定义域	domain	/dəʊˈmeɪn/	The range of values for the independent variable of a function
连续性	continuity	/kɒntɪˈnjuːəti/	The property that a function has no jumps or breaks at a point

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