A 'hole' in the graph that can often be 'removed' by simplifying the function (e.g., factoring). The limit exists, but the function is not defined at that point, or defined to a different value.

A sudden 'jump' in the graph where the left and right-hand limits are different. Common in piecewise functions.

The function approaches infinity (or negative infinity) as x approaches a certain value, creating a vertical asymptote.

They often involve algebraic manipulation (factoring) to identify and remove them, a skill frequently tested.

Almost exclusively in piecewise functions, where the different pieces of the function do not connect.

The function's values increase or decrease without bound as x approaches a specific value.

Look for breaks, holes, jumps, or vertical asymptotes in the graph.

For a function to be continuous at a point, the limit must exist at that point and be equal to the function's value.

Discontinuities affect the behavior of functions and are crucial for understanding limits, derivatives, and integrals.

Factoring can reveal common factors that lead to removable discontinuities in rational functions.

1. Factor the numerator and denominator. 2. Cancel common factors. 3. The cancelled factor indicates the x-value of the discontinuity. 4. Evaluate the simplified function at that x-value to find the y-value of the 'hole'.

1. Factor and simplify the function. 2. Substitute the x-value of the discontinuity into the simplified function.

1. Identify the x-value where the function changes definition. 2. Find the left-hand limit at that x-value. 3. Find the right-hand limit at that x-value. 4. If the limits are different, there's a jump discontinuity.

1. Check if the function is defined at the point. 2. Find the left-hand limit. 3. Find the right-hand limit. 4. If the limits exist and are equal to the function's value at that point, it's continuous.

1. Look for values of x that make the denominator of a rational function equal to zero. 2. Check the limits as x approaches these values from both sides. 3. If the limits approach infinity (or negative infinity), there's an asymptote discontinuity.

1. Identify the discontinuity. 2. Find the limit at that point. 3. Define a new piecewise function that equals the original function everywhere except at the discontinuity, where it equals the limit.

1. Set the pieces of the function equal to each other at the point where they meet. 2. Solve for the unknown variable.

A function cannot be differentiable at a point of discontinuity.

1. Rewrite the absolute value function as a piecewise function. 2. Analyze the continuity of the resulting piecewise function.

1. Set the denominator of the rational function equal to zero. 2. Solve for x. 3. The solution(s) are the equations of the vertical asymptotes.

Removable: Limit exists, point missing. | Jump: Left and right-hand limits differ.

Removable: Limit exists, point missing. | Asymptote: Function approaches infinity.

Jump: Finite difference between left and right limits. | Asymptote: Function approaches infinity.

Continuous: No breaks, limits exist and equal function values. | Discontinuous: Has breaks, holes, or jumps.

Left-hand: Limit as x approaches from the left. | Right-hand: Limit as x approaches from the right.

Piecewise: Defined by different functions over different intervals. | Rational: Ratio of two polynomials.

Removable: Limit exists and is finite. | Asymptote: Limit is infinite or does not exist.

Undefined: Function simply has no value at that point. | Removable: Limit exists even though the function is undefined or has a different value.

Jump: Graph has a clear vertical jump. | Asymptote: Graph approaches a vertical line infinitely closely.

Hole: A single point is missing. | Vertical Asymptote: The function approaches infinity.