Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

1. Limits and Continuity

Continuity

Calculus

1. Limits and Continuity

Continuity

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2:29 minutes

Problem 2.5.4

Textbook Question

In Exercises 1–4, say whether the function graphed is continuous on [−1, 3]. If not, where does it fail to be continuous and why?
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Verified step by step guidance

Step 1: Understand the definition of continuity. A function is continuous on an interval if it is continuous at every point in that interval. A function is continuous at a point if the limit of the function as it approaches the point from both sides is equal to the function's value at that point.

Step 2: Identify the interval of interest, which is [-1, 3]. This means we need to check the continuity of the function at every point within this interval, including the endpoints -1 and 3.

Step 3: Examine the graph of the function over the interval [-1, 3]. Look for any breaks, jumps, or holes in the graph, as these indicate points where the function may not be continuous.

Step 4: Check the endpoints of the interval. Ensure that the function is defined at x = -1 and x = 3, and that the limits from the left and right at these points match the function's value at these points.

Step 5: Identify any points within the interval where the function is not continuous. This could be due to a jump discontinuity, an infinite discontinuity, or a removable discontinuity. Note these points and explain why the function fails to be continuous at each one.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Continuity of Functions

A function is continuous on an interval if there are no breaks, jumps, or holes in its graph within that interval. Formally, a function f(x) is continuous at a point c if the limit of f(x) as x approaches c equals f(c). Understanding this concept is crucial for determining whether a function is continuous over a specified range.

Types of Discontinuities

Discontinuities can be classified into three main types: removable, jump, and infinite. A removable discontinuity occurs when a function is not defined at a point but can be made continuous by redefining it. A jump discontinuity happens when the left-hand and right-hand limits at a point do not match, while an infinite discontinuity occurs when the function approaches infinity at a point. Identifying these types helps in pinpointing where a function fails to be continuous.

Evaluating Limits

Limits are fundamental in analyzing the behavior of functions as they approach specific points. To determine continuity, one must evaluate the limit of the function at the endpoints and any critical points within the interval. If the limit exists and equals the function's value at those points, the function is continuous there. This concept is essential for assessing continuity over a given interval.