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Continuity

Emily Davis

Emily Davis

6 min read

Next Topic - Intermediate Value Theorem

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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

#Table of Contents

  1. Introduction to Continuity
  2. Polynomial Functions
  3. Rational Functions
  4. Exponential Functions
  5. Logarithmic Functions
  6. Trigonometric Functions
  7. Practice Questions
  8. Glossary
  9. 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 xxx, along with possible constant terms.

#Examples

- Polynomial functions: - f(x)=x2−6x+15f(x) = x^2 - 6x + 15f(x)=x2−6x+15 - g(x)=x5−x4+3x2+xg(x) = x^5 - x^4 + 3x^2 + xg(x)=x5−x4+3x2+x
  • Non-polynomial functions:
    • h(x)=x3−1xh(x) = x^3 - \frac{1}{x}h(x)=x3−x1​
      • Because 1x=x−1\frac{1}{x} = x^{-1}x1​=x−1 is not a positive integer power of xxx.
    • j(x)=x2+x−7j(x) = x^2 + \sqrt{x} - 7j(x)=x2+x​−7
      • Because x\sqrt{x}x​ is not a positive integer power of xxx.

#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

- Rational function: - x4+2x2−3x−7\frac{x^4 + 2x^2 - 3}{x - 7}x−7x4+2x2−3​

#Continuity

Key Concept

Rational functions are continuous everywhere, except at points where their denominator is zero.

- x2+7x2−x−2=x2+7(x+1)(x−2)\frac{x^2 + 7}{x^2 - x - 2} = \frac{x^2 + 7}{(x + 1)(x - 2)}x2−x−2x2+7​=(x+1)(x−2)x2+7​ - This function is **not continuous** at x=−1x = -1x=−1 or x=2x = 2x=2. - It is continuous over all other real numbers or any smaller domain excluding x=−1x = -1x=−1 or x=2x = 2x=2.

The graphs of rational functions have vertical asymptotes at points where their denominators are zero.


#Exponential Functions

#Definition

An exponential function is a function of the form kepxke^{px}kepx or kapxka^{px}kapx, where kkk and ppp are non-zero real number constants, aaa is a positive real number constant, and eee is the exponential constant (approximately 2.7182).

#Example

- f(x)=5e3xf(x) = 5e^{3x}f(x)=5e3x - g(x)=2⋅34xg(x) = 2 \cdot 3^{4x}g(x)=2⋅34x

#Continuity

Key Concept

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

The graphs of exponential functions have horizontal asymptotes at y=0y = 0y=0.


#Logarithmic Functions

#Definition

A logarithmic function is a function of the form kln⁡(px)k\ln(px)kln(px) or klog⁡a(px)k\log_a(px)kloga​(px), where kkk and ppp are non-zero real number constants, and aaa is a positive real number constant.

#Example

- f(x)=ln⁡(2x)f(x) = \ln(2x)f(x)=ln(2x) - g(x)=log⁡_3(5x)g(x) = \log\_3(5x)g(x)=log_3(5x)

#Continuity

Key Concept

Logarithmic functions are continuous at all points where px>0px > 0px>0.

- ln⁡(2x)\ln(2x)ln(2x) is continuous for all x>0x > 0x>0. - ln⁡(−3x)\ln(-3x)ln(−3x) is continuous for all x<0x < 0x<0.

The graphs of logarithmic functions have vertical asymptotes at x=0x = 0x=0.


#Trigonometric Functions

#Sine and Cosine Functions

#Definition

Sine and cosine functions of the form ksin⁡(px)k \sin(px)ksin(px) or kcos⁡(px)k \cos(px)kcos(px), where kkk and ppp are non-zero real number constants.

#Continuity

Key Concept

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

#Tangent Functions

#Definition

A tangent function of the form ktan⁡(px)k \tan(px)ktan(px), where kkk and ppp are non-zero real number constants.

#Continuity

Key Concept

Tangent functions are continuous at all points where pxpxpx is not an odd multiple of π2\frac{\pi}{2}2π​.

- ktan⁡(px)k \tan(px)ktan(px) is continuous at all points where px≠…,−5π2,−3π2,−π2,π2,3π2,5π2,…px \ne \ldots, -\frac{5\pi}{2}, -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldotspx=…,−25π​,−23π​,−2π​,2π​,23π​,25π​,… - At points where px=…,−5π2,−3π2,−π2,π2,3π2,5π2,…px = \ldots, -\frac{5\pi}{2}, -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldotspx=…,−25π​,−23π​,−2π​,2π​,23π​,25π​,…, tangent functions have **vertical asymptotes**.

#Practice Questions

Practice Question

#Multiple Choice Questions

  1. Which of the following functions is a polynomial function?

    • A) f(x)=x3+1xf(x) = x^3 + \frac{1}{x}f(x)=x3+x1​
    • B) g(x)=x4−4x2+2g(x) = x^4 - 4x^2 + 2g(x)=x4−4x2+2
    • C) h(x)=2x+3xh(x) = 2^x + 3xh(x)=2x+3x
    • D) j(x)=ln⁡(x)+x2j(x) = \ln(x) + x^2j(x)=ln(x)+x2

    Answer: B

  2. At which points is the function f(x)=x2+1x−3f(x) = \frac{x^2 + 1}{x - 3}f(x)=x−3x2+1​ not continuous?

    • A) x=0x = 0x=0
    • B) x=3x = 3x=3
    • C) x=−1x = -1x=−1
    • D) x=1x = 1x=1

    Answer: B

#Short Answer Questions

  1. Explain why the function h(x)=x2+x−7h(x) = x^2 + \sqrt{x} - 7h(x)=x2+x​−7 is not a polynomial function.

  2. Determine the points of discontinuity for the function f(x)=x2−4x2−x−6f(x) = \frac{x^2 - 4}{x^2 - x - 6}f(x)=x2−x−6x2−4​.


#Glossary

  • Continuous Function: A function without breaks, jumps, or holes in its domain.
  • Polynomial Function: A function consisting of sums or differences of positive integer powers of xxx.
  • Rational Function: A function defined as the quotient of two polynomials.
  • Exponential Function: A function where the variable xxx appears as an exponent.
  • Logarithmic Function: A function where the variable xxx appears inside a logarithm.
  • Trigonometric Functions: Functions of angles, including sine, cosine, and tangent functions.

#Summary and Key Takeaways

#Summary

  • Polynomial functions are continuous on all points of their domains.
  • Rational functions are continuous everywhere except where their denominators are zero.
  • Exponential functions are continuous on all points of their domains.
  • Logarithmic functions are continuous where px>0px > 0px>0.
  • Trigonometric functions are continuous within specific intervals.

#Key Takeaways

  • Understand the types of functions and their continuity.
  • Polynomial and exponential functions are generally continuous over all real numbers.
  • Rational functions have points of discontinuity at zero denominators.
  • Logarithmic functions are continuous where the argument is positive.
  • Trigonometric functions have specific intervals of continuity.

Exam Tip

When preparing for exams, focus on understanding the conditions for continuity of different types of functions and practice identifying points of discontinuity.

Common Mistake

Students often confuse the conditions for continuity of rational and logarithmic functions. Ensure you understand that rational functions are discontinuous at zero denominators, while logarithmic functions are discontinuous where their arguments are non-positive.

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Question 1 of 10

Which of the following functions is a polynomial function? 😎

f(x)=x3+1xf(x) = x^3 + \frac{1}{x}f(x)=x3+x1​

g(x)=x4−4x2+2g(x) = x^4 - 4x^2 + 2g(x)=x4−4x2+2

h(x)=2x+3xh(x) = 2^x + 3xh(x)=2x+3x

j(x)=ln⁡(x)+x2j(x) = \ln(x) + x^2j(x)=ln(x)+x2