Glossary

Continuity on an Interval

Criticality: 3

A function is continuous on an interval if it is continuous at every single point within that interval, meaning its graph can be traced without lifting a pencil.

Example:

The function $f(x) = e^x$ is continuous on an interval $(-\infty, \infty)$ because its graph has no breaks or jumps anywhere.

Differentiability

Criticality: 3

The property of a function having a well-defined derivative at a point or over an interval, implying the function is smooth and has no sharp corners or vertical tangents.

Example:

While continuity is necessary for differentiability, a function like $f(x) = |x|$ is continuous at $x=0$ but not differentiable due to a sharp corner.

Discontinuities

Criticality: 3

Points or intervals where a function is not continuous, often characterized by holes, jumps, or vertical asymptotes.

Example:

A function like $f(x) = \frac{1}{x}$ has a discontinuity at $x=0$ because it has a vertical asymptote there.

Domain Restrictions

Criticality: 2

Conditions that limit the set of input values for which a function is defined, often arising from operations like square roots or division by zero.

Example:

When analyzing $f(x) = \frac{1}{\sqrt{x-5}}$ , you must consider domain restrictions to ensure $x-5 > 0$ , meaning $x > 5$ .

Function Value

Criticality: 3

The output of a function at a specific input point, denoted as $f(a)$.

Example:

For a function to be continuous at $x=a$ , its function value $f(a)$ must exist and be equal to both the left-hand and right-hand limits at $x=a$ .

Left-hand limit

Criticality: 3

The value a function approaches as the input variable approaches a specific point from values less than that point.

Example:

For a jump discontinuity at $x=c$ , the left-hand limit $lim_{x\to c^{-}} f(x)$ will not equal the right-hand limit.

Piecewise Functions

Criticality: 3

Functions defined by multiple sub-functions, each applicable over a certain interval of the domain.

Example:

To check if a piecewise function like $f(x) = {x^2 \text{ for } x<0, x+1 \text{ for } x \ge 0}$ is continuous, you must examine the connection point at $x=0$ .

Rational Functions

Criticality: 2

Functions expressed as a ratio of two polynomials, where the denominator cannot be zero.

Example:

The rational function $g(x) = \frac{x-1}{x^2-9}$ has vertical asymptotes and discontinuities where its denominator, $x^2-9$ , equals zero.

Right-hand limit

Criticality: 3

The value a function approaches as the input variable approaches a specific point from values greater than that point.

Example:

To confirm continuity at $x=5$ , you must check if the right-hand limit $lim_{x\to 5^{+}} f(x)$ matches the left-hand limit and the function value.

Square Roots

Criticality: 2

A mathematical operation $\sqrt{z}$ that requires the radicand $z$ to be non-negative ($z \geq 0$) to yield real numbers.

Example:

For $f(x) = \sqrt{x^2 - 4}$ , the square root imposes a domain restriction where $x^2 - 4 \ge 0$ , leading to $x \in (-\infty, -2] \cup [2, \infty)$ .

Glossary

Continuity on an Interval

Criticality: 3

A function is continuous on an interval if it is continuous at every single point within that interval, meaning its graph can be traced without lifting a pencil.

Example:

The function $f(x) = e^x$ is continuous on an interval $(-\infty, \infty)$ because its graph has no breaks or jumps anywhere.

Differentiability

Criticality: 3

The property of a function having a well-defined derivative at a point or over an interval, implying the function is smooth and has no sharp corners or vertical tangents.

Example:

While continuity is necessary for differentiability, a function like $f(x) = |x|$ is continuous at $x=0$ but not differentiable due to a sharp corner.

Discontinuities

Criticality: 3

Points or intervals where a function is not continuous, often characterized by holes, jumps, or vertical asymptotes.

Example:

A function like $f(x) = \frac{1}{x}$ has a discontinuity at $x=0$ because it has a vertical asymptote there.

Domain Restrictions

Criticality: 2

Conditions that limit the set of input values for which a function is defined, often arising from operations like square roots or division by zero.

Example:

When analyzing $f(x) = \frac{1}{\sqrt{x-5}}$ , you must consider domain restrictions to ensure $x-5 > 0$ , meaning $x > 5$ .

Function Value

Criticality: 3

The output of a function at a specific input point, denoted as $f(a)$.

Example:

For a function to be continuous at $x=a$ , its function value $f(a)$ must exist and be equal to both the left-hand and right-hand limits at $x=a$ .

Left-hand limit

Criticality: 3

The value a function approaches as the input variable approaches a specific point from values less than that point.

Example:

For a jump discontinuity at $x=c$ , the left-hand limit $lim_{x\to c^{-}} f(x)$ will not equal the right-hand limit.

Piecewise Functions

Criticality: 3

Functions defined by multiple sub-functions, each applicable over a certain interval of the domain.

Example:

To check if a piecewise function like $f(x) = {x^2 \text{ for } x<0, x+1 \text{ for } x \ge 0}$ is continuous, you must examine the connection point at $x=0$ .

Rational Functions

Criticality: 2

Functions expressed as a ratio of two polynomials, where the denominator cannot be zero.

Example:

The rational function $g(x) = \frac{x-1}{x^2-9}$ has vertical asymptotes and discontinuities where its denominator, $x^2-9$ , equals zero.

Right-hand limit

Criticality: 3

The value a function approaches as the input variable approaches a specific point from values greater than that point.

Example:

To confirm continuity at $x=5$ , you must check if the right-hand limit $lim_{x\to 5^{+}} f(x)$ matches the left-hand limit and the function value.

Square Roots

Criticality: 2

A mathematical operation $\sqrt{z}$ that requires the radicand $z$ to be non-negative ($z \geq 0$) to yield real numbers.

Example:

For $f(x) = \sqrt{x^2 - 4}$ , the square root imposes a domain restriction where $x^2 - 4 \ge 0$ , leading to $x \in (-\infty, -2] \cup [2, \infty)$ .