Calculus Continuity and Limits Vocabulary

Flashcards covering key vocabulary terms, definitions, and rules related to continuity, types of discontinuities, continuous intervals, and function domains from Chapter 1 through Chapter 8.

Continuous Function

A function where the limit as $x$ approaches $a$ is equal to the function value $f(a)$, requiring that there are no holes, the limit exists, and no division by zero occurs.

Function Value ($f(a)$)

The $y$ value of a function when $x$ equals $a$, represented visually on a graph by a solid dot.

Removable Discontinuity

A type of discontinuity that occurs when a rational function has a common factor that factors out, resulting in a line with a hole.

Nonremovable Discontinuity

A type of discontinuity, such as a jump discontinuity or a vertical asymptote, where the left-hand limit does not equal the right-hand limit.

Jump Discontinuity

A nonremovable discontinuity characterized by a gap on the graph where the left-hand limit and right-hand limit equal two different $y$ values; it takes priority over a hole if both exist at the same point.

Continuous from the Right

A condition where the limit of $f(x)$ as $x$ approaches $a$ from the right equals $f(a)$, represented by a solid dot when approaching from the right side.

Continuous from the Left

A condition where the limit of $f(x)$ as $x$ approaches $a$ from the left equals $f(a)$, represented by a solid dot when approaching from the left side.

Continuous on an Interval

A property of a function where it is continuous at every individual number within a given interval.

Domain of a Polynomial Function

All real numbers, written as the interval $(-\infty, \infty)$, on which polynomial functions are continuous everywhere.

Domain of a Rational Function

The set of all real numbers except those that make the denominator equal to zero.

Radicand

The mathematical expression located underneath a square root or radical symbol.

Domain of a Radical Function

The set of values found by setting the radicand greater than or equal to zero ($\ge 0$) and solving for $x$.

Composite Function

A function formed by placing an inside function $g(x)$ inside an outside function $f(x)$, written as $f(g(x))$.