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
#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.
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 , along with possible constant terms.
#Examples
- Non-polynomial functions:
-
- Because is not a positive integer power of .
-
- Because is not a positive integer power of .
-
#Continuity
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
#Continuity
Rational functions are continuous everywhere, except at points where their denominator is zero.
#Exponential Functions
#Definition
An exponential function is a function of the form or , where and are non-zero real number constants, is a positive real number constant, and is the exponential constant (approximately 2.7182).
#Example
#Continuity
Exponential functions are continuous at all points in their domains, which include all real numbers or any smaller defined domain.
#Logarithmic Functions
#Definition
A logarithmic function is a function of the form or , where and are non-zero real number constants, and is a positive real number constant.
#Example
#Continuity
Logarithmic functions are continuous at all points where .
#Trigonometric Functions
#Sine and Cosine Functions
#Definition
Sine and cosine functions of the form or , where and are non-zero real number constants.
#Continuity
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 , where and are non-zero real number constants.
#Continuity
Tangent functions are continuous at all points where is not an odd multiple of .
#Practice Questions
Practice Question
#Multiple Choice Questions
-
Which of the following functions is a polynomial function?
- A)
- B)
- C)
- D)
Answer: B
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At which points is the function not continuous?
- A)
- B)
- C)
- D)
Answer: B
#Short Answer Questions
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Explain why the function is not a polynomial function.
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Determine the points of discontinuity for the function .
#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 .
- Rational Function: A function defined as the quotient of two polynomials.
- Exponential Function: A function where the variable appears as an exponent.
- Logarithmic Function: A function where the variable 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 .
- 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.
When preparing for exams, focus on understanding the conditions for continuity of different types of functions and practice identifying points of discontinuity.
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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