Glossary

Continuous Function

Criticality: 3

A function whose graph can be drawn without lifting the pencil, meaning there are no breaks, jumps, or holes in its domain.

Example:

The polynomial function $f(x) = x^3 - 2x + 1$ is a continuous function for all real numbers, as its graph is a smooth, unbroken curve.

Discontinuities

Criticality: 3

Points where a function is not continuous, appearing as gaps, jumps, or holes in its graph.

Example:

The function $f(x) = 1/x$ has a discontinuity at $x=0$ because it's undefined there, creating a vertical asymptote.

Left-hand limit

Criticality: 2

The value a function approaches as the input approaches a certain point from values less than (to the left of) that point.

Example:

To check if a piecewise function is continuous at $x=3$ , you would evaluate the left-hand limit by using the function definition for $x < 3$ .

Limit

Criticality: 3

The value that a function approaches as the input (x) approaches a certain point, regardless of the function's actual value at that point.

Example:

For $f(x) = (x^2 - 1)/(x - 1)$ , the limit as $x$ approaches 1 is 2, even though the function is undefined at $x=1$ .

Piecewise Functions

Criticality: 2

Functions defined by multiple sub-functions, each applying to a different interval of the domain.

Example:

A common example is a cell phone plan where the cost per minute changes after a certain number of minutes used, forming a piecewise function.

Removable Discontinuities

Criticality: 3

A type of discontinuity where the limit of a function exists at a point, but the function is either undefined or has a different value at that point, creating a 'hole' that can be 'filled' by redefining the function.

Example:

The function $f(x) = (x^2 - 4)/(x - 2)$ has a removable discontinuity at $x=2$ because it simplifies to $x+2$ for $x eq 2$ , indicating a hole at $(2,4)$ .

Right-hand limit

Criticality: 2

The value a function approaches as the input approaches a certain point from values greater than (to the right of) that point.

Example:

When analyzing a function at $x=5$ , the right-hand limit is found by considering values of $x$ slightly larger than 5.