Glossary

Continuous

Criticality: 3

A function is continuous at a point if its graph can be drawn without lifting the pencil, meaning the limit exists, the function is defined, and the limit equals the function's value at that point.

Example:

The function $f(x) = x^2$ is continuous everywhere, as you can draw its parabola without any breaks.

Differentiable

Criticality: 3

A function is differentiable at a point if its derivative exists at that point, implying the function is smooth and has a well-defined tangent line.

Example:

The function $f(x) = \sin(x)$ is differentiable at every point, as its graph is always smooth and has a unique tangent line.

Discontinuities

Criticality: 3

Points where a function is not continuous, characterized by breaks, holes, or asymptotes in the graph, which always imply non-differentiability.

Example:

The function $f(x) = 1/x$ has a discontinuity at $x=0$ due to a vertical asymptote.

Infinite Discontinuity

Criticality: 2

A type of discontinuity where the function's value approaches positive or negative infinity as x approaches a certain point, typically associated with vertical asymptotes.

Example:

The function $f(x) = 1/(x-2)^2$ has an infinite discontinuity at $x=2$ , where the graph shoots up towards infinity.

Jump Discontinuity

Criticality: 2

A type of discontinuity where the function's value 'jumps' from one value to another at a specific point, meaning the left-hand and right-hand limits are finite but unequal.

Example:

A piecewise function defined as $f(x) = {x \text{ for } x<0, x+2 \text{ for } x \ge 0}$ exhibits a jump discontinuity at $x=0$ .

Left-hand limit of the derivative

Criticality: 2

The value that the derivative of a function approaches as x approaches a specific point from the left side.

Example:

For a piecewise function, evaluating the derivative of the left-defined piece at the boundary point gives the left-hand limit of the derivative.

Removable Discontinuity

Criticality: 2

A type of discontinuity where a single point is missing or misplaced, creating a 'hole' in the graph, but the limit of the function exists at that point.

Example:

The function $f(x) = (x^2-1)/(x-1)$ has a removable discontinuity at $x=1$ , as it simplifies to $x+1$ with a hole at $x=1$ .

Right-hand limit of the derivative

Criticality: 2

The value that the derivative of a function approaches as x approaches a specific point from the right side.

Example:

For a piecewise function, evaluating the derivative of the right-defined piece at the boundary point gives the right-hand limit of the derivative.

Sharp Turns (Corners or Cusps)

Criticality: 3

Points on a graph where the direction changes abruptly, making the function non-differentiable because the left and right derivatives do not match.

Example:

The absolute value function $f(x) = |x|$ has a sharp turn (a corner) at $x=0$ , where its derivative is undefined.

Smooth

Criticality: 2

A characteristic of a function's graph at a point where it is differentiable, meaning there are no abrupt changes in direction or breaks.

Example:

The curve of $y = x^3$ is smooth at $x=0$ , allowing for a clear tangent line, unlike a sharp corner.

Vertical Tangents

Criticality: 2

Points on a graph where the tangent line is vertical, meaning the slope is undefined (the derivative approaches infinity or negative infinity), rendering the function non-differentiable.

Example:

The function $f(x) = x^{1/3}$ has a vertical tangent at $x=0$ , where its derivative is undefined.

Glossary

Continuous

Criticality: 3

A function is continuous at a point if its graph can be drawn without lifting the pencil, meaning the limit exists, the function is defined, and the limit equals the function's value at that point.

Example:

The function $f(x) = x^2$ is continuous everywhere, as you can draw its parabola without any breaks.

Differentiable

Criticality: 3

A function is differentiable at a point if its derivative exists at that point, implying the function is smooth and has a well-defined tangent line.

Example:

The function $f(x) = \sin(x)$ is differentiable at every point, as its graph is always smooth and has a unique tangent line.

Discontinuities

Criticality: 3

Points where a function is not continuous, characterized by breaks, holes, or asymptotes in the graph, which always imply non-differentiability.

Example:

The function $f(x) = 1/x$ has a discontinuity at $x=0$ due to a vertical asymptote.

Infinite Discontinuity

Criticality: 2

A type of discontinuity where the function's value approaches positive or negative infinity as x approaches a certain point, typically associated with vertical asymptotes.

Example:

The function $f(x) = 1/(x-2)^2$ has an infinite discontinuity at $x=2$ , where the graph shoots up towards infinity.

Jump Discontinuity

Criticality: 2

A type of discontinuity where the function's value 'jumps' from one value to another at a specific point, meaning the left-hand and right-hand limits are finite but unequal.

Example:

A piecewise function defined as $f(x) = {x \text{ for } x<0, x+2 \text{ for } x \ge 0}$ exhibits a jump discontinuity at $x=0$ .

Left-hand limit of the derivative

Criticality: 2

The value that the derivative of a function approaches as x approaches a specific point from the left side.

Example:

For a piecewise function, evaluating the derivative of the left-defined piece at the boundary point gives the left-hand limit of the derivative.

Removable Discontinuity

Criticality: 2

A type of discontinuity where a single point is missing or misplaced, creating a 'hole' in the graph, but the limit of the function exists at that point.

Example:

The function $f(x) = (x^2-1)/(x-1)$ has a removable discontinuity at $x=1$ , as it simplifies to $x+1$ with a hole at $x=1$ .

Right-hand limit of the derivative

Criticality: 2

The value that the derivative of a function approaches as x approaches a specific point from the right side.

Example:

For a piecewise function, evaluating the derivative of the right-defined piece at the boundary point gives the right-hand limit of the derivative.

Sharp Turns (Corners or Cusps)

Criticality: 3

Points on a graph where the direction changes abruptly, making the function non-differentiable because the left and right derivatives do not match.

Example:

The absolute value function $f(x) = |x|$ has a sharp turn (a corner) at $x=0$ , where its derivative is undefined.

Smooth

Criticality: 2

A characteristic of a function's graph at a point where it is differentiable, meaning there are no abrupt changes in direction or breaks.

Example:

The curve of $y = x^3$ is smooth at $x=0$ , allowing for a clear tangent line, unlike a sharp corner.

Vertical Tangents

Criticality: 2

Points on a graph where the tangent line is vertical, meaning the slope is undefined (the derivative approaches infinity or negative infinity), rendering the function non-differentiable.

Example:

The function $f(x) = x^{1/3}$ has a vertical tangent at $x=0$ , where its derivative is undefined.