#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

- Rational function: -

\frac{x^4 + 2x^2 - 3}{x - 7}

#Continuity

Key Concept

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

-

\frac{x^2 + 7}{x^2 - x - 2} = \frac{x^2 + 7}{(x + 1)(x - 2)}

- This function is **not continuous** at

x = -1

or

x = 2

. - It is continuous over all other real numbers or any smaller domain excluding

x = -1

or

x = 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 $ke^{px}$ or $ka^{px}$ , where $k$ and $p$ are non-zero real number constants, $a$ is a positive real number constant, and $e$ is the exponential constant (approximately 2.7182).

#Example

-

f(x) = 5e^{3x}

-

g(x) = 2 \cdot 3^{4x}

#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 $y = 0$ .

#Logarithmic Functions

#Definition

A logarithmic function is a function of the form $k\ln(px)$ or $k\log_a(px)$ , where $k$ and $p$ are non-zero real number constants, and $a$ is a positive real number constant.

#Example

-

f(x) = \ln(2x)

-

g(x) = \log\_3(5x)

#Continuity

Key Concept

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

-

\ln(2x)

is continuous for all

x > 0

. -

\ln(-3x)

is continuous for all

x < 0

.

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

#Trigonometric Functions

#Sine and Cosine Functions

#Definition

Sine and cosine functions of the form $k \sin(px)$ or $k \cos(px)$ , where $k$ and $p$ 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 $k \tan(px)$ , where $k$ and $p$ are non-zero real number constants.

#Continuity

Key Concept

Tangent functions are continuous at all points where $px$ is not an odd multiple of $\frac{\pi}{2}$ .

-

k \tan(px)

is continuous at all points where

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}, \ldots

- At points where

px = \ldots, -\frac{5\pi}{2}, -\frac{3\pi}{2}, -\frac{\pi}{2}, \frac{\pi}{2}, \frac{3\pi}{2}, \frac{5\pi}{2}, \ldots

, tangent functions have **vertical asymptotes**.

#Practice Questions

Practice Question

#Multiple Choice Questions

Which of the following functions is a polynomial function?
- A) $f(x) = x^3 + \frac{1}{x}$
- B) $g(x) = x^4 - 4x^2 + 2$
- C) $h(x) = 2^x + 3x$
- D) $j(x) = \ln(x) + x^2$
Answer: B
At which points is the function $f(x) = \frac{x^2 + 1}{x - 3}$ not continuous?
- A) $x = 0$
- B) $x = 3$
- C) $x = -1$
- D) $x = 1$
Answer: B

#Short Answer Questions

Explain why the function $h(x) = x^2 + \sqrt{x} - 7$ is not a polynomial function.
Determine the points of discontinuity for the function $f(x) = \frac{x^2 - 4}{x^2 - x - 6}$ .

#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 $x$ .
Rational Function: A function defined as the quotient of two polynomials.
Exponential Function: A function where the variable $x$ appears as an exponent.
Logarithmic Function: A function where the variable $x$ 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 $px > 0$ .
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.