/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 3 What does it mean for a function... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 2: Problem 3

What does it mean for a function to be continuous on an interval?

Short Answer

Expert verified
Answer: A function is continuous on an interval if its limit as x approaches any point in the interval exists and equals the function's value at that point, meaning there are no gaps or jumps in the function. Some properties of continuous functions include the sum, difference, product, and quotient of two continuous functions being continuous, their graphs appearing unbroken, and their compliance with the Intermediate Value Theorem. Examples of continuous functions are polynomial functions, trigonometric functions, and exponential and logarithmic functions.

Step by step solution

01

Definition of Continuity

A function f(x) is said to be continuous on an interval [a, b] if for all x in [a, b], the limit of f(x) as x approaches c exists and is equal to f(c) for all c in [a, b]. In mathematical terms, this is expressed as: Limit as x approaches c of f(x) = f(c)
02

Properties of Continuous Functions

Continuous functions have several properties, such as: 1. The sum, difference, product and quotient of two continuous functions are continuous. 2. Continuous functions are unbroken and have no gaps or jumps in their graphs. 3. The Intermediate Value Theorem applies to continuous functions, which states that if f(x) is continuous on the interval [a, b], and there exists a number k between f(a) and f(b), then there exists a number c in (a, b) such that f(c) = k.
03

Examples of Continuous Functions

Some examples of continuous functions are: 1. Polynomial functions: These functions have no gaps or jumps and are continuous over their entire domain. 2. Trigonometric functions: Functions like sine and cosine are continuous over their entire domain. 3. Exponential and logarithmic functions: These functions are continuous over their entire domain. Overall, a function is considered continuous on an interval if it does not have any gaps or jumps, and its limit as x approaches any point in the interval exists and equals the function's value at that point.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the limit of the following sequences or state that the limit does not exist. $$\begin{aligned} &\left\\{4,2, \frac{4}{3}, 1, \frac{4}{5}, \frac{2}{3}, \ldots\right\\}, \text { which is defined by } f(n)=\frac{4}{n}, \text { for }\\\ &n=1,2,3, \ldots \end{aligned}$$

Let \(f(x)=\frac{2 e^{x}+10 e^{-x}}{e^{x}+e^{-x}} .\) Analyze \(\lim _{x \rightarrow 0} f(x), \lim _{x \rightarrow-\infty} f(x),\) and \(\lim _{x \rightarrow \infty} f(x) .\) Then give the horizontal and vertical asymptotes of \(f .\) Plot \(f\) to verify your results.

Limits by graphing Use the zoom and trace features of a graphing utility to approximate the following limits. $$\lim _{x \rightarrow 1} \frac{18(\sqrt[3]{x}-1)}{x^{3}-1}$$

The Heaviside function is used in engineering applications to model flipping a switch. It is defined as $$H(x)=\left\\{\begin{array}{ll} 0 & \text { if } x<0 \\ 1 & \text { if } x \geq 0 \end{array}\right.$$ a. Sketch a graph of \(H\) on the interval [-1,1] b. Does \(\lim _{x \rightarrow 0} H(x)\) exist? Explain your reasoning after first examining \(\lim _{x \rightarrow 0^{-}} H(x)\) and \(\lim _{x \rightarrow 0^{+}} H(x)\)

The magnitude of the electric field at a point \(x\) meters from the midpoint of a \(0.1-\mathrm{m}\) line of charge is given by \(E(x)=\frac{4.35}{x \sqrt{x^{2}+0.01}}(\text { in units of newtons per coulomb }, \mathrm{N} / \mathrm{C}).\) Evaluate \(\lim _{x \rightarrow 10} E(x)\).
See all solutions

Recommended explanations on Math Textbooks

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Probability and Statistics

Read Explanation

Discrete Mathematics

Read Explanation

Logic and Functions

Read Explanation

Geometry

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.