Continuity

Emily Davis

6 min read

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Study Guide Overview

This study guide covers the continuity of common functions including polynomial, rational, exponential, logarithmic, and trigonometric functions. It defines each function type, explains their continuity conditions, and provides practice questions. Key takeaways include identifying points of discontinuity for rational functions (zero denominators) and logarithmic functions (non-positive arguments).

#Continuity of Common Functions

#Introduction to Continuity

Proving that a given function is continuous at all points in its domain can be quite tedious. Luckily, there are standard results for many commonly occurring types of functions.

Exam Tip

Understand and apply the concept of continuity to various types of functions.

#Polynomial Functions

#Definition

A polynomial function is a function composed of sums or differences of positive integer powers of $x$ , along with possible constant terms.

#Examples

- Polynomial functions: -

f(x) = x^2 - 6x + 15

g(x) = x^5 - x^4 + 3x^2 + x

Non-polynomial functions:
- $h(x) = x^3 - \frac{1}{x}$
  - Because $\frac{1}{x} = x^{-1}$ is not a positive integer power of $x$ .
- $j(x) = x^2 + \sqrt{x} - 7$
  - Because $\sqrt{x}$ is not a positive integer power of $x$ .

#Continuity

Key Concept

Polynomial functions are continuous at all points in their domains, which include all real numbers or any smaller defined domain.

#Rational Functions

#Definition

A rational function is a function defined as a fraction (or quotient) where both the numerator and denominator are polynomials.

#Examples

<exampl...

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Previous Topic - Non-removable Discontinuities Next Topic - Intermediate Value Theorem

Continuity

Emily Davis

6 min read

Next Topic - Intermediate Value Theorem

Listen to this study note

Study Guide Overview

#Continuity of Common Functions

#Table of Contents

Introduction to Continuity
Polynomial Functions
Rational Functions
Exponential Functions
Logarithmic Functions
Trigonometric Functions
Practice Questions
Glossary
Summary and Key Takeaways

#Introduction to Continuity

Proving that a given function is continuous at all points in its domain can be quite tedious. Luckily, there are standard results for many commonly occurring types of functions.

Exam Tip

Understand and apply the concept of continuity to various types of functions.

#Polynomial Functions

#Definition

A polynomial function is a function composed of sums or differences of positive integer powers of $x$ , along with possible constant terms.

#Examples

- Polynomial functions: -

f(x) = x^2 - 6x + 15

g(x) = x^5 - x^4 + 3x^2 + x

Non-polynomial functions:
- $h(x) = x^3 - \frac{1}{x}$
  - Because $\frac{1}{x} = x^{-1}$ is not a positive integer power of $x$ .
- $j(x) = x^2 + \sqrt{x} - 7$
  - Because $\sqrt{x}$ is not a positive integer power of $x$ .

#Continuity

Key Concept

Polynomial functions are continuous at all points in their domains, which include all real numbers or any smaller defined domain.

#Rational Functions

#Definition

A rational function is a function defined as a fraction (or quotient) where both the numerator and denominator are polynomials.

#Examples

<exampl...

How are we doing?

Give us your feedback and let us know how we can improve

Previous Topic - Non-removable Discontinuities Next Topic - Intermediate Value Theorem