AP Calculus AB Unit 1 (Limits & Continuity): Continuity, Discontinuities, IVT, and How to “Fix” Holes

Continuous function

A function is continuous at a point if the limit equals the function value at that point.

Definition of Continuity Checklist

The three conditions for a function to be continuous at x = c: function value exists, limit exists, and limit equals function value.

Removable discontinuity

A type of discontinuity where the limit exists and is finite, but the function value is either undefined or does not equal the limit.

Jump discontinuity

A type of discontinuity where the left-hand limit and right-hand limit exist but are not equal.

Infinite discontinuity

A type of discontinuity that occurs when a function approaches infinity as x approaches a certain value.

Intermediate Value Theorem (IVT)

If a function is continuous on [a,b], then it takes every value between f(a) and f(b) at some point in [a,b].

Continuous on an open interval

A function is continuous on (a,b) if it is continuous at every point between a and b.

Continuous on a closed interval

A function is continuous on [a,b] if it is continuous on (a,b), continuous from the right at a, and continuous from the left at b.

Limit of a function

The value that a function approaches as the input approaches a particular point.

Piecewise function

A function defined by multiple sub-functions, each applying to a specific interval.

Two-sided limit

A limit that exists only if the left-hand and right-hand limits are equal.

Left-hand limit

The value that a function approaches as the input approaches a certain point from the left.

Right-hand limit

The value that a function approaches as the input approaches a certain point from the right.

Polynomials

Types of functions that are continuous for all real numbers.

Rational functions

Types of functions that are continuous wherever the denominator is non-zero.

Root functions

Functions like square roots that are continuous where they are defined (for real outputs).

Vertical asymptote

A line where a function approaches infinity or negative infinity.

Oscillating discontinuity

A discontinuity where a function oscillates infinitely fast so the limit does not exist.

Function value exists

Condition 1 for continuity indicating that the function has a defined value at c.

The limit exists

Condition 2 for continuity indicating that both one-sided limits exist and are finite.

Limit equals value

Condition 3 for continuity indicating that the limit of the function as x approaches c equals f(c).

Domain restrictions

Values that input into a function which cause it to be undefined.

Finding constants for continuity

Using limits and values to determine parameters that make a function continuous at certain points.

Common mistake of continuity

Assuming a function is continuous only because f(c) exists without verifying limits.

Algebraic approach to limit

Using algebraic manipulation, often to resolve indeterminate forms like 0/0.

Undeterminate form 0/0

Signals the need to factor and simplify, often indicating a removable discontinuity.

Substitution in limits

Direct replacement of x with c in a continuous function should yield f(c).