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Glossary

C

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)=exf(x) = e^x is continuous on an interval (−∞,∞)(-\infty, \infty) because its graph has no breaks or jumps anywhere.

D

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∣f(x) = |x| is continuous at x=0x=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)=1xf(x) = \frac{1}{x} has a discontinuity at x=0x=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)=1x−5f(x) = \frac{1}{\sqrt{x-5}}, you must consider domain restrictions to ensure x−5>0x-5 > 0, meaning x>5x > 5.

F

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

L

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=cx=c, the left-hand limit limx→c−f(x)lim_{x\to c^{-}} f(x) will not equal the right-hand limit.

P

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)=x2 for x<0,x+1 for x≥0f(x) = {x^2 \text{ for } x<0, x+1 \text{ for } x \ge 0} is continuous, you must examine the connection point at x=0x=0.

R

Rational Functions

Criticality: 2

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

Example:

The rational function g(x)=x−1x2−9g(x) = \frac{x-1}{x^2-9} has vertical asymptotes and discontinuities where its denominator, x2−9x^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=5x=5, you must check if the right-hand limit limx→5+f(x)lim_{x\to 5^{+}} f(x) matches the left-hand limit and the function value.

S

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)=x2−4f(x) = \sqrt{x^2 - 4}, the square root imposes a domain restriction where x2−4≥0x^2 - 4 \ge 0, leading to x∈(−∞,−2]∪[2,∞)x \in (-\infty, -2] \cup [2, \infty).