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Questions and Answers
What are the conditions for a function f(x) to be continuous at a point a?
What are the conditions for a function f(x) to be continuous at a point a?
What is one way to identify if a function is continuous?
What is one way to identify if a function is continuous?
What property of continuity states that the sum of two continuous functions is continuous?
What property of continuity states that the sum of two continuous functions is continuous?
What is NOT a method to identify continuity of a function?
What is NOT a method to identify continuity of a function?
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What is the purpose of limit analysis in identifying continuity?
What is the purpose of limit analysis in identifying continuity?
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What is the result of the composition of two continuous functions?
What is the result of the composition of two continuous functions?
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What is a characteristic of a continuous function at a point?
What is a characteristic of a continuous function at a point?
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What is true about the product of two continuous functions?
What is true about the product of two continuous functions?
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How many conditions are required for a function to be continuous at a point?
How many conditions are required for a function to be continuous at a point?
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What can be used to identify discontinuities in a function?
What can be used to identify discontinuities in a function?
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Study Notes
Function Continuity
- A function
f(x)is said to be continuous at a pointaif the following three conditions are satisfied:-
f(a)is defined -
lim x→a f(x)exists -
lim x→a f(x) = f(a)
-
- In other words, a function is continuous at a point if the limit of the function as
xapproachesais equal to the value of the function ata.
Identifying Continuity of a Function
- A function can be identified as continuous if it satisfies the following properties:
- The function is defined at the point: The function has a value at the point in question.
-
The function has a limit at the point: The limit of the function as
xapproaches the point exists. -
The limit equals the function value: The limit of the function as
xapproaches the point is equal to the value of the function at that point.
- Methods to identify continuity include:
- Graphical analysis: Visual inspection of the graph to check for gaps, holes, or jumps.
-
Limit analysis: Calculating the limit of the function as
xapproaches the point to check for existence and equality. - Algebraic analysis: Simplifying the function and checking for discontinuities.
Properties of Continuity
- Sum Rule: The sum of two continuous functions is continuous.
- Product Rule: The product of two continuous functions is continuous.
- Chain Rule: The composition of two continuous functions is continuous.
- Inverse Rule: The inverse of a continuous function is continuous, if it exists.
-
Intermediate Value Theorem: If a function is continuous on a closed interval and takes on values
f(a)andf(b)at the endpoints, then it takes on all values betweenf(a)andf(b)at some point in the interval. - Extreme Value Theorem: A continuous function on a closed interval takes on a maximum and minimum value on that interval.
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Description
Learn about the definition and properties of continuous functions, including the sum, product, chain, and inverse rules, as well as the intermediate value and extreme value theorems.
