AP Calculus BC Flashcards: Removing Discontinuities
Written by AP Content Team, Verified for 2026 AP Exams, Last updated: May 2026
Review key ideas with interactive flashcards. This set includes 10 cards to help you master important concepts.
What does it mean to 'redefine the value of a function' to remove a discontinuity?
It means assigning a new output value to the function at the specific x-value of the discontinuity, making the function's value equal to its limit at that point.
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What does it mean to 'redefine the value of a function' to remove a discontinuity?
It means assigning a new output value to the function at the specific x-value of the discontinuity, making the function's value equal to its limit at that point.
How do you remove a discontinuity at a point where the limit exists?
You remove the discontinuity by defining or redefining the value of the function at that point to be equal to the value of the limit as x approaches that point.
For a piecewise-defined function, what must be true for it to be continuous at a boundary point?
The value of the expression defining the function on one side of the boundary must equal the value of the expression on the other side, and both must equal the function's value at the boundary.
What is the key condition that must be met to remove a discontinuity at a point?
The limit of the function must exist at the point of discontinuity. If the limit does not exist, the discontinuity cannot be removed.
Can all types of discontinuities be removed by redefining a single point?
No, only removable discontinuities, where the limit exists, can be fixed this way. Jump or infinite discontinuities cannot be removed by redefining a single point.
What is a removable discontinuity?
A removable discontinuity is a point on a function's graph where the limit exists, but the function is not continuous. It can be 'removed' by redefining the function's value at that specific point.
What is the relationship between a function's limit and its value for ensuring continuity at a point?
For a function to be continuous at a point, the limit as x approaches that point must exist, the function must be defined at that point, and these two values must be equal.
If a function f(x) has a removable discontinuity at x=5 and the limit as x approaches 5 is 10, what value should f(5) be redefined to?
To make the function continuous at x=5, the value of f(5) should be redefined to be 10, which is the value of the limit.
For a piecewise function to be continuous at a boundary, what three specific values must all be equal?
The limit from the left, the limit from the right, and the value of the function at the boundary point must all be equal to each other.
What is the first step in solving for a parameter that makes a piecewise function continuous at a boundary?
Set the expression for the function on the left side of the boundary equal to the expression for the function on the right side of the boundary at that specific boundary point.