AP Calculus BC Flashcards: Exploring Types of 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 is the fundamental process for justifying a conclusion about continuity at a specific point?
To justify a conclusion about continuity at a point, one must use the formal definition of continuity, which involves checking if the function's value and limit exist and are equal at that point.
Card 1 of 10
All Flashcards (10)
What is the fundamental process for justifying a conclusion about continuity at a specific point?
To justify a conclusion about continuity at a point, one must use the formal definition of continuity, which involves checking if the function's value and limit exist and are equal at that point.
Define a Removable Discontinuity.
A removable discontinuity is a type of discontinuity where the limit of the function exists at a point, but the function is either undefined or has a different value at that point, creating a 'hole' in the graph.
If a function fails the first condition of the definition of continuity at a point (i.e., f(c) is undefined), which types of discontinuities are possible?
If f(c) is undefined, the discontinuity could be either removable (if the limit exists) or due to a vertical asymptote (if the limit is infinite).
Define a Discontinuity due to a Vertical Asymptote.
A discontinuity due to a vertical asymptote occurs at a point where the function's value increases or decreases without bound (approaches ±∞) as x approaches that point from at least one side.
Which type of discontinuity is also known as an infinite discontinuity?
A discontinuity due to a vertical asymptote is also referred to as an infinite discontinuity because the function's values approach infinity at that point.
What are the three types of discontinuities mentioned in the content?
The three types of discontinuities are removable discontinuities, jump discontinuities, and discontinuities due to vertical asymptotes.
Define a Jump Discontinuity.
A jump discontinuity occurs at a point where the function's left-hand limit and right-hand limit both exist but are not equal to each other, causing a sudden 'jump' in the graph.
If the limit of a function as x approaches 'a' exists, but f(a) is undefined, what type of discontinuity is present?
This describes a removable discontinuity, as the function could be made continuous at that point by defining f(a) to be equal to the limit.
What three conditions must be met for a function f(x) to be continuous at a point x=c, according to the definition of continuity?
For a function to be continuous at x=c: 1) f(c) must be defined, 2) the limit of f(x) as x approaches c must exist, and 3) the limit must equal the function's value, lim f(x) = f(c).
A function's graph approaches y=2 from the left of x=5 and y=-1 from the right of x=5. What type of discontinuity occurs at x=5?
This describes a jump discontinuity because the left-hand limit (2) and the right-hand limit (-1) exist but are not equal.