Glossary

Asymptote Discontinuity

Criticality: 2

A type of discontinuity where the function's limits approach positive or negative infinity as x approaches a certain value. This creates a vertical asymptote that the graph approaches but never touches.

Example:

The function $f(x) = 1/(x-2)$ has an asymptote discontinuity at $x=2$ , where the graph shoots up or down infinitely.

Continuity

Criticality: 3

A property of a function where its graph can be drawn without lifting the pencil. For a function to be continuous at a point, it must be defined, its limit must exist, and the limit must equal the function's value at that point.

Example:

A polynomial function like $f(x) = x^3 - 2x + 5$ demonstrates continuity across its entire domain, as there are no breaks or jumps.

Differentiable

Criticality: 3

A property of a function where its derivative exists at a given point, implying the function is continuous and has a well-defined, non-vertical tangent line at that point.

Example:

The function $f(x) = x^2$ is differentiable at every point, but $f(x) = |x|$ is not differentiable at $x=0$ because of the sharp corner.

Discontinuity

Criticality: 3

A point where a function is not continuous, meaning its graph cannot be drawn without lifting the pencil. It occurs when one or more of the conditions for continuity are not met.

Example:

The function $f(x) = 1/x$ has a discontinuity at $x=0$ because it is undefined there and the graph breaks.

Jump Discontinuity

Criticality: 2

A type of discontinuity where the function's value suddenly 'jumps' from one value to another. This occurs when the left-hand limit does not equal the right-hand limit at a specific point.

Example:

The greatest integer function, $f(x) = \lfloor x \rfloor$ , exhibits a jump discontinuity at every integer value, like at $x=2$ where it jumps from 1 to 2.

Left-hand Limit

Criticality: 2

The value that a function approaches as the input x gets closer and closer to a specific point from values less than that point.

Example:

For the piecewise function $f(x) = {x+1 \text{ if } x<0; x-1 \text{ if } x \ge 0}$ , the left-hand limit as $x$ approaches 0 is 1.

Limit

Criticality: 3

The value that a function 'approaches' as the input (x) gets arbitrarily close to a certain point. It is a fundamental concept for defining continuity and derivatives.

Example:

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

Piecewise Function

Criticality: 3

A function defined by multiple sub-functions, each applying to a different interval of the domain. These are frequently used to illustrate different types of discontinuities.

Example:

The function $f(x) = {x^2 \text{ if } x<1; x+1 \text{ if } x \ge 1}$ is a piecewise function that might have a discontinuity at $x=1$ .

Rational Function

Criticality: 2

A function that can be expressed as the ratio of two polynomial functions, where the denominator is not zero. These functions are common sources of removable and asymptote discontinuities.

Example:

The function $f(x) = (x^2 + 1)/(x - 5)$ is a rational function that has a vertical asymptote at $x=5$ .

Removable Discontinuity

Criticality: 3

A type of discontinuity where a single point is missing or out of place, often appearing as a 'hole' in the graph. It can often be 'removed' by redefining the function at that single point.

Example:

The function $f(x) = (x^2 - 9)/(x - 3)$ has a removable discontinuity at $x=3$ because the factor $(x-3)$ cancels out, leaving a hole.

Right-hand Limit

Criticality: 2

The value that a function approaches as the input x gets closer and closer to a specific point from values greater than that point.

Example:

For the piecewise function $f(x) = {x+1 \text{ if } x<0; x-1 \text{ if } x \ge 0}$ , the right-hand limit as $x$ approaches 0 is -1.

Glossary

Asymptote Discontinuity

Criticality: 2

Example:

The function $f(x) = 1/(x-2)$ has an asymptote discontinuity at $x=2$ , where the graph shoots up or down infinitely.

Continuity

Criticality: 3

Example:

A polynomial function like $f(x) = x^3 - 2x + 5$ demonstrates continuity across its entire domain, as there are no breaks or jumps.

Differentiable

Criticality: 3

A property of a function where its derivative exists at a given point, implying the function is continuous and has a well-defined, non-vertical tangent line at that point.

Example:

The function $f(x) = x^2$ is differentiable at every point, but $f(x) = |x|$ is not differentiable at $x=0$ because of the sharp corner.

Discontinuity

Criticality: 3

A point where a function is not continuous, meaning its graph cannot be drawn without lifting the pencil. It occurs when one or more of the conditions for continuity are not met.

Example:

The function $f(x) = 1/x$ has a discontinuity at $x=0$ because it is undefined there and the graph breaks.

Jump Discontinuity

Criticality: 2

A type of discontinuity where the function's value suddenly 'jumps' from one value to another. This occurs when the left-hand limit does not equal the right-hand limit at a specific point.

Example:

The greatest integer function, $f(x) = \lfloor x \rfloor$ , exhibits a jump discontinuity at every integer value, like at $x=2$ where it jumps from 1 to 2.

Left-hand Limit

Criticality: 2

The value that a function approaches as the input x gets closer and closer to a specific point from values less than that point.

Example:

For the piecewise function $f(x) = {x+1 \text{ if } x<0; x-1 \text{ if } x \ge 0}$ , the left-hand limit as $x$ approaches 0 is 1.

Limit

Criticality: 3

The value that a function 'approaches' as the input (x) gets arbitrarily close to a certain point. It is a fundamental concept for defining continuity and derivatives.

Example:

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

Piecewise Function

Criticality: 3

A function defined by multiple sub-functions, each applying to a different interval of the domain. These are frequently used to illustrate different types of discontinuities.

Example:

The function $f(x) = {x^2 \text{ if } x<1; x+1 \text{ if } x \ge 1}$ is a piecewise function that might have a discontinuity at $x=1$ .

Rational Function

Criticality: 2

A function that can be expressed as the ratio of two polynomial functions, where the denominator is not zero. These functions are common sources of removable and asymptote discontinuities.

Example:

The function $f(x) = (x^2 + 1)/(x - 5)$ is a rational function that has a vertical asymptote at $x=5$ .

Removable Discontinuity

Criticality: 3

A type of discontinuity where a single point is missing or out of place, often appearing as a 'hole' in the graph. It can often be 'removed' by redefining the function at that single point.

Example:

The function $f(x) = (x^2 - 9)/(x - 3)$ has a removable discontinuity at $x=3$ because the factor $(x-3)$ cancels out, leaving a hole.

Right-hand Limit

Criticality: 2

The value that a function approaches as the input x gets closer and closer to a specific point from values greater than that point.

Example:

For the piecewise function $f(x) = {x+1 \text{ if } x<0; x-1 \text{ if } x \ge 0}$ , the right-hand limit as $x$ approaches 0 is -1.