Continuity

Emily Davis

5 min read

Next Topic - Non-removable Discontinuities

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Study Guide Overview

This guide covers removable discontinuities, focusing on their definition, identification, and removal. It explains how to determine if a discontinuity is removable by evaluating limits. The guide provides worked examples, practice questions, and exam strategies for handling these types of problems. Key terms include continuous function, limit, and removable discontinuity.

#Study Notes on Removable Discontinuity

#Introduction to Removable Discontinuity

Understanding removable discontinuities is crucial for mastering continuity in functions, a fundamental topic in calculus. This guide will help you identify and remove these discontinuities, ensuring a smooth learning experience.

#Definition and Concept

#What is a Removable Discontinuity?

A removable discontinuity is a type of discontinuity in a function that can be "removed" to make the function continuous over an interval that includes the discontinuity.

Key Concept

A function $f$ is continuous at the point $x=c$ if:

f(c)

exists, -

\lim\_{{x \to c}} f(x)

exists, - and

\lim\_{{x \to c}} f(x) = f(c)

#Characteristics of Removable Discontinuity

A removable discontinuity is essentially a "hole" in the function. It is a point where:

$\lim_{{x \to c}} f(x)$ exists,
but $...

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Previous Topic - Continuity Next Topic - Non-removable Discontinuities

Question 1 of 8

A function $f$ is continuous at $x=c$ if which of the following conditions are met? 🤔

$f(c)$ exists, $\lim_{x \to c} f(x)$ exists, and $\lim_{x \to c} f(x) = f(c)$

$f(c)$ does not exist, $\lim_{x \to c} f(x)$ exists, and $\lim_{x \to c} f(x) \neq f(c)$

$\lim_{x \to c} f(x)$ exists, but $f(c)$ does not exist

$f(c)$ exists, but $\lim_{x \to c} f(x)$ does not exist