Continuity of Polynomial Functions

A polynomial function is the most basic type of elementary function and plays an important role in calculus. Understanding the continuity properties of polynomial functions is essential for studying more complicated types of function.

Basic Properties

A polynomial function has the following basic properties:

• Domain: $\mathbb{R}$ (all real numbers)
• Continuity: continuous everywhere on its domain
• Graph features: a smooth curve, with no jumps or breaks
• Differentiability: differentiable everywhere on its domain

Definition of Polynomial Functions

Continuity of Polynomial Functions

Basic Theorem

Proof Strategy

The proof of the continuity of polynomial functions rests on the following facts:

1. Constant functions are continuous: $f(x) = c$ is continuous on $\mathbb{R}$
2. Power functions are continuous: $f(x) = x^n$ is continuous on $\mathbb{R}$
3. The sum, difference and product of continuous functions are continuous

Detailed Proof

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

1. Every power function $x^k$ is continuous on $\mathbb{R}$
2. Every constant multiple $a_k x^k$ is continuous on $\mathbb{R}$
3. The sum of finitely many continuous functions is continuous on $\mathbb{R}$
4. Therefore the polynomial function is continuous on $\mathbb{R}$

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
• Its slope is $a$
• It is 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
• It opens upwards when $a > 0$ and downwards when $a < 0$
• It is 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
• It may have local extrema
• It is continuous everywhere on $\mathbb{R}$

Exercises

Exercise 1

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

Exercise 2

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

Exercise 3

Let $f(x) = x^4 - 2x^3 + x^2 - 1$. Find the intervals of continuity of $f(x)$.

Exercise 4

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

Exercise 5

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

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Summary

Symbols Used in This Article