Glossary

Continuity

Criticality: 3

A property of a function where its graph can be drawn without lifting the pen, meaning it behaves smoothly without sudden jumps or gaps at a given point or interval.

Example:

A polynomial function like $f(x) = x^3 - 2x + 1$ is continuous for all real numbers, as its graph flows smoothly without any breaks.

Discontinuity

Criticality: 3

A point or interval where a function is not continuous, indicating a break, jump, or hole in its graph.

Example:

The function $f(x) = 1/(x-1)$ has a discontinuity at $x=1$ because the function is undefined there, leading to a vertical asymptote.

Domain

Criticality: 1

The set of all possible input values (x-values) for which a function is defined and produces a real output.

Example:

The domain of the function $f(x) = \sqrt{x-4}$ is $x \geq 4$ , because the expression under the square root must be non-negative.

Function value (f(c))

Criticality: 3

The specific output value of a function when a particular input 'c' from its domain is substituted into the function.

Example:

For the function $f(x) = x^2 + 5$ , the function value at $x=2$ is $f(2) = 2^2 + 5 = 9$ .

Jump discontinuity

Criticality: 2

A type of discontinuity where the left-hand limit and the right-hand limit at a point both exist but are not equal, causing a sudden 'jump' in the graph.

Example:

The greatest integer function, $f(x) = \lfloor x floor$ , has a jump discontinuity at every integer value, as the function value abruptly changes.

Left-hand limit

Criticality: 2

The value a function approaches as the input (x) gets arbitrarily close to a specific point 'c' from values less than 'c'.

Example:

For a piecewise function $f(x) = x+1$ for $x<2$ , the left-hand limit as x approaches 2 is $2+1=3$ .

Limit of a function

Criticality: 3

The value that a function approaches as its input (x) gets arbitrarily close to a specific point 'c', regardless of the function's actual value at 'c'.

Example:

For $f(x) = (x^2-9)/(x-3)$ , the limit of the function as x approaches 3 is 6, even though the function is undefined at $x=3$ .

Removable discontinuity

Criticality: 2

A type of discontinuity where the limit of the function exists at a point, but the function value is either undefined or does not match the limit, often appearing as a 'hole' in the graph.

Example:

The function $f(x) = (x^2-1)/(x-1)$ has a removable discontinuity at $x=1$ because it can be simplified to $x+1$ for $x eq 1$ , creating a hole at $(1,2)$ .

Right-hand limit

Criticality: 2

The value a function approaches as the input (x) gets arbitrarily close to a specific point 'c' from values greater than 'c'.

Example:

For a piecewise function $f(x) = 2x-1$ for $x \geq 2$ , the right-hand limit as x approaches 2 is $2(2)-1=3$ .

Vertical asymptote

Criticality: 2

A vertical line $x=c$ that the graph of a function approaches but never touches, typically occurring where the function's denominator is zero and the numerator is non-zero.

Example:

The function $f(x) = an(x)$ has vertical asymptotes at $x = \pi/2 + n\pi$ (where n is an integer), as the function's value approaches infinity or negative infinity at these points.

Glossary

Continuity

Criticality: 3

A property of a function where its graph can be drawn without lifting the pen, meaning it behaves smoothly without sudden jumps or gaps at a given point or interval.

Example:

A polynomial function like $f(x) = x^3 - 2x + 1$ is continuous for all real numbers, as its graph flows smoothly without any breaks.

Discontinuity

Criticality: 3

A point or interval where a function is not continuous, indicating a break, jump, or hole in its graph.

Example:

The function $f(x) = 1/(x-1)$ has a discontinuity at $x=1$ because the function is undefined there, leading to a vertical asymptote.

Domain

Criticality: 1

The set of all possible input values (x-values) for which a function is defined and produces a real output.

Example:

The domain of the function $f(x) = \sqrt{x-4}$ is $x \geq 4$ , because the expression under the square root must be non-negative.

Function value (f(c))

Criticality: 3

The specific output value of a function when a particular input 'c' from its domain is substituted into the function.

Example:

For the function $f(x) = x^2 + 5$ , the function value at $x=2$ is $f(2) = 2^2 + 5 = 9$ .

Jump discontinuity

Criticality: 2

A type of discontinuity where the left-hand limit and the right-hand limit at a point both exist but are not equal, causing a sudden 'jump' in the graph.

Example:

The greatest integer function, $f(x) = \lfloor x floor$ , has a jump discontinuity at every integer value, as the function value abruptly changes.

Left-hand limit

Criticality: 2

The value a function approaches as the input (x) gets arbitrarily close to a specific point 'c' from values less than 'c'.

Example:

For a piecewise function $f(x) = x+1$ for $x<2$ , the left-hand limit as x approaches 2 is $2+1=3$ .

Limit of a function

Criticality: 3

The value that a function approaches as its input (x) gets arbitrarily close to a specific point 'c', regardless of the function's actual value at 'c'.

Example:

For $f(x) = (x^2-9)/(x-3)$ , the limit of the function as x approaches 3 is 6, even though the function is undefined at $x=3$ .

Removable discontinuity

Criticality: 2

A type of discontinuity where the limit of the function exists at a point, but the function value is either undefined or does not match the limit, often appearing as a 'hole' in the graph.

Example:

The function $f(x) = (x^2-1)/(x-1)$ has a removable discontinuity at $x=1$ because it can be simplified to $x+1$ for $x eq 1$ , creating a hole at $(1,2)$ .

Right-hand limit

Criticality: 2

The value a function approaches as the input (x) gets arbitrarily close to a specific point 'c' from values greater than 'c'.

Example:

For a piecewise function $f(x) = 2x-1$ for $x \geq 2$ , the right-hand limit as x approaches 2 is $2(2)-1=3$ .

Vertical asymptote

Criticality: 2

A vertical line $x=c$ that the graph of a function approaches but never touches, typically occurring where the function's denominator is zero and the numerator is non-zero.

Example:

The function $f(x) = an(x)$ has vertical asymptotes at $x = \pi/2 + n\pi$ (where n is an integer), as the function's value approaches infinity or negative infinity at these points.