Textbook Question
For what values of a and b is
g(x) = { ax + 2b, x ≤ 0
x² + 3a – b, 0 < x ≤ 2
3x – 5, x > 2
continuous at every x?
1. Limits and Continuity
Continuity
2:30 minutesProblem 2.5.60
Textbook Question
If functions f(x) and g(x) are continuous for 0 ≤ x ≤ 1, could f(x)/g(x) possibly be discontinuous at a point of [0,1]? Give reasons for your answer.
Verified step by step guidance1
To determine if f(x)/g(x) could be discontinuous, we need to consider the conditions under which a quotient of two continuous functions might be discontinuous.
Recall that a function is discontinuous at a point if it is not defined or if the limit does not exist or does not equal the function's value at that point.
For f(x)/g(x) to be discontinuous, g(x) must be zero at some point in the interval [0,1] because division by zero is undefined.
Even if g(x) is zero at a point, f(x)/g(x) could still be continuous if the limit of f(x)/g(x) as x approaches that point exists and equals the value of the function at that point.
Therefore, f(x)/g(x) could be discontinuous at a point in [0,1] if g(x) is zero at that point and the limit of f(x)/g(x) as x approaches that point does not exist or does not equal the function's value at that point.
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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 at a point if the limit of the function as it approaches that point equals the function's value at that point. For functions f(x) and g(x) to be continuous on the interval [0, 1], they must not have any breaks, jumps, or asymptotes within that range. This property is crucial for understanding how the behavior of these functions affects their quotient.
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Quotient of Functions
The quotient of two functions, f(x)/g(x), is defined wherever g(x) is not equal to zero. If g(x) approaches zero at any point in the interval [0, 1], the quotient may become undefined or exhibit discontinuity at that point. Thus, the behavior of g(x) is critical in determining the continuity of the quotient function.
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Discontinuity in Quotients
A function can be discontinuous if it involves division by a function that is zero at some point. Even if f(x) and g(x) are continuous, if g(x) equals zero at a point in [0, 1], then f(x)/g(x) will be discontinuous at that point. This highlights the importance of analyzing the denominator in the context of continuity.
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