AP Calculus AB Flashcards: Defining Continuity at a Point
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.
Why is it insufficient to only check if f(c) exists to determine continuity at c?
A function can have a value at f(c) but still have a jump or hole at that point, meaning the limit either doesn't exist or doesn't equal the function's value.
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Why is it insufficient to only check if f(c) exists to determine continuity at c?
A function can have a value at f(c) but still have a jump or hole at that point, meaning the limit either doesn't exist or doesn't equal the function's value.
What is the primary tool used to justify conclusions about continuity at a point?
The primary tool is the three-part definition of continuity, which requires checking the existence of the function's value, the existence of the limit, and their equality.
To justify that a function is continuous at a point, what three specific pieces of information must you establish and compare?
You must establish the value of the limit from the left, the limit from the right, and the value of the function at the point, then show they are all equal.
How does the final condition, lim (x→c) f(x) = f(c), connect the concepts of limits and function values?
This condition ensures that the value a function approaches near a point is the same as the function's actual value at that point, effectively 'plugging the hole' in the graph.
State the formal definition of continuity at a point x=c in a single equation.
A function f is continuous at x=c provided that lim (x→c) f(x) = f(c). This single equation implicitly covers all three conditions.
If you know that lim (x→c) f(x) exists and f(c) exists, but the function is not continuous at c, which condition must have failed?
The final condition must have failed: the limit of f(x) as x approaches c is not equal to the function's value, f(c). This describes a removable discontinuity (a hole).
What does the existence of lim (x→c) f(x) imply about the function's behavior around x=c?
It implies that the function approaches the same finite value from both the left and the right side of c, preventing a jump discontinuity.
A function has a jump discontinuity at x=2. Which of the three conditions for continuity must fail?
The second condition, that the limit of f(x) as x approaches 2 exists, must fail because the left-hand and right-hand limits are not equal.
In the definition of continuity, what does the condition 'f(c) exists' ensure?
This condition ensures that the point x=c is in the domain of the function, meaning the function has a defined value at that specific point.
What are the three conditions required for a function f to be continuous at a point x=c?
The three conditions are: 1) f(c) exists, 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).