Determine the value(s) of (if any) for which the function is discontinuous.
Table of contents
- 0. Functions4h 53m
- 1. Limits and Continuity2h 2m
- 2. Intro to Derivatives1h 33m
- 3. Techniques of Differentiation2h 18m
- 4. Derivatives of Exponential & Logarithmic Functions1h 16m
- 5. Applications of Derivatives2h 19m
- 6. Graphical Applications of Derivatives6h 0m
- 7. Antiderivatives & Indefinite Integrals48m
- 8. Definite Integrals4h 36m
- 9. Graphical Applications of Integrals1h 43m
- 10. Integrals of Inverse, Exponential, & Logarithmic Functions21m
- 11. Techniques of Integration2h 7m
- 12. Trigonometric Functions6h 54m
- Angles29m
- Trigonometric Functions on Right Triangles1h 8m
- Solving Right Triangles23m
- Trigonometric Functions on the Unit Circle1h 19m
- Graphs of Sine & Cosine46m
- Graphs of Other Trigonometric Functions32m
- Trigonometric Identities52m
- Derivatives of Trig Functions42m
- Integrals of Basic Trig Functions28m
- Integrals of Other Trig Functions10m
- 13: Intro to Differential Equations2h 23m
- 14. Sequences & Series2h 8m
- 15. Power Series2h 19m
- 16. Probability & Calculus45m
1. Limits and Continuity
Continuity
Multiple Choice
Use the graph of f(x) to determine if the function is continuous or discontinuous at x=c.
c=3

A
Continuous
B
Discontinuous
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Verified step by step guidance1
Step 1: Recall the definition of continuity. A function is continuous at a point x = c if the following three conditions are satisfied: (1) f(c) is defined, (2) the limit of f(x) as x approaches c exists, and (3) the limit of f(x) as x approaches c is equal to f(c).
Step 2: Analyze the graph at x = 3. Observe that there is a break in the graph at x = 3, indicating a potential discontinuity.
Step 3: Check if f(3) is defined. From the graph, there is no point at x = 3 on the curve, which means f(3) is not defined.
Step 4: Evaluate the limit of f(x) as x approaches 3. The graph shows that the left-hand limit and right-hand limit do not converge to the same value, indicating the limit does not exist.
Step 5: Conclude that the function is discontinuous at x = 3 because it fails the conditions for continuity: f(3) is not defined, and the limit of f(x) as x approaches 3 does not exist.
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