Recent Lessons

Show all results for ""

Feature Overview

Ace your exams with our all-in-one platform for creating and sharing quizzes and tests.

Quizzes

Create quizzes and tests automatically from your content using AI.

Flashcards

Automatically turn your notes into digital flashcards.

Share, Export & Embed

Share with classmates or export to Excel and your learning management system.

Stats & Reporting

Auto-grading quizzes and tests with detailed stats and reports.

Mobile Apps

The smarter way to study – wherever you are.

Pricing Schools Business

Features Pricing Schools Business

5 Questions

0 Views

Continuity at a Point - Calculus

Choose a study mode

Play Quiz

Study Flashcards

Spaced Repetition

Chat to Lesson

Podcast

Play an AI-generated podcast conversation about this lesson

Questions and Answers

What condition must be satisfied for a function $f$ to be continuous at a point $c$?

$ ext{CHL} eq ext{RHL}$

$ ext{RHL} = f(c)$

$ ext{CHL} = 0$

$ ext{CHL} = ext{RHL} = f(c)$ (correct)

Which of the following correctly describes the right-hand limit of a function at a point $c$?

$ ext{RHL} = ext{CHL}$

$ ext{RHL} = rac{f(c)}{c}$

$ ext{RHL} = ext{lim}_{x o c^{+}}f(x)$ (correct)

$ ext{RHL} = f(c - ext{small})$

Which of the following statements about continuity is incorrect?

For continuity, both limits must equal $f(c)$.

If $ ext{CHL} eq ext{RHL}$, $f$ cannot be continuous at $c$.

A function can be continuous even if $ ext{CHL} = f(c)$.

Continuity can occur at a point where $f(c)$ is undefined. (correct)

What does the term 'left-hand limit' refer to in relation to a function at point $c$?

<p>$ ext{lim}_{x o c^{-}}f(x)$ (D)</p> Signup and view all the answers

What can be concluded if $ ext{CHL} = f(c)$ and $ ext{RHL} eq f(c)$?

<p>The function is not continuous at $c$. (A)</p> Signup and view all the answers

Flashcards

Continuity at a Point

A function is continuous at a point if the limit of the function as x approaches the point is equal to the function's value at the point.

Left-hand Limit

The limit of a function as x approaches a point from the left-hand side.

Right-hand Limit

The limit of a function as x approaches a point from the right-hand side.

Continuity Requirements

A function is continuous at a point if the left-hand limit, right-hand limit, and the function's value at that point are all equal.

Signup and view all the flashcards

CHL = RHL = f(c)

For a function to be continuous at a point, the left-hand limit must equal the right-hand limit, which in turn must equal the function's value at that point.

Signup and view all the flashcards

Study Notes

Continuity at a Point

A function, f, is continuous at a point c if the limit of f(x) as x approaches c is equal to f(c).
This can be broken down into two parts: the left-hand limit and the right-hand limit.
Left-hand limit (LHL) : lim (x→c⁻) f(x)
Right-hand limit (RHL): lim (x→c⁺) f(x)
For a function to be continuous at a point c, the limit from the left and the limit from the right must exist and be equal to the function's value at c, and f(c) must be defined.
Mathematically, this is written as:
- LHL = RHL = f(c)

Studying That Suits You

Use AI to generate personalized quizzes and flashcards to suit your learning preferences.

Description

This quiz covers the concept of continuity at a specific point in a function. It includes definitions of left-hand limit and right-hand limit, and the criteria for a function being continuous at a point. Familiarize yourself with the mathematical expressions and concepts involved in continuity.