AP Calculus AB Flashcards: Defining Continuity At A Point

What this deck covers

This deck focuses on Defining Continuity At A Point, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus AB.

How to use these flashcards

Work through these flashcards in short sessions. Try to answer each prompt before flipping the card, then revisit any cards you miss until the explanation feels automatic.

Flashcard 1: State the definition of a removable discontinuity.

Answer: A point where $f(x)$ is not defined or $\text{lim}_{x \to c} f(x) \neq f(c)$, but can be redefined. The gap can be filled by redefining the function at that point.

Flashcard 2: Verify continuity of $f(x) = \frac{1}{x^2}$ at $x = 0$.

Answer: $f(x)$ is not continuous at $x = 0$. Function undefined at $x = 0$ (division by zero).

Flashcard 3: Find if $f(x) = \frac{x - 1}{x^2 - 1}$ is continuous at $x = 1$.

Answer: Removable discontinuity at $x = 1$. Factor gives $\text{lim}_{x \to 1} \frac{1}{x+1} = \frac{1}{2}$ but $f(1)$ undefined.

Flashcard 4: What is the third condition for continuity at a point $x = c$?

Answer: $\lim_{x \to c} f(x) = f(c)$. The limit value must equal the function value.

Flashcard 5: What is a continuous function?

Answer: A function that is continuous at every point in its domain. No breaks, holes, or jumps anywhere in the domain.

Flashcard 6: Determine if $f(x) = \text{sin}(x)$ is continuous for all $x$.

Answer: $f(x)$ is continuous for all $x$. Trigonometric functions are continuous on their domains.

Flashcard 7: What is the conclusion if $\text{lim}_{x \to c} f(x) = f(c)$ and $f(c)$ is defined?

Answer: The function $f(x)$ is continuous at $x = c$. All three conditions for continuity are satisfied.

Flashcard 8: Which condition fails if $\text{lim}_{x \to c} f(x)$ does not exist?

Answer: The condition that $\text{lim}_{x \to c} f(x)$ exists. The left and right limits don't agree.

Flashcard 9: What is the first condition for continuity at a point $x = c$?

Answer: $f(c)$ is defined. The function must have a value at point $c$.

Flashcard 10: Determine if $f(x) = \frac{1}{x}$ is continuous for $x \neq 0$.

Answer: $f(x)$ is continuous for $x \neq 0$. Rational functions are continuous except where denominator is zero.

Flashcard 11: State the definition of a jump discontinuity.

Answer: A point where left and right limits exist but are not equal. The function jumps from one value to another at that point.

Flashcard 12: State the Intermediate Value Theorem (IVT) condition for continuity.

Answer: If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then $\text{exists } c \text{ in } (a, b)$ with $f(c) = N$. Continuous functions take on all intermediate values.

Flashcard 13: State whether $f(x) = \frac{\text{sin}(x)}{x}$ is continuous at $x = 0$.

Answer: Removable discontinuity at $x = 0$. $\text{lim}_{x \to 0} \frac{\text{sin}(x)}{x} = 1$ but $f(0)$ undefined.

Flashcard 14: Evaluate continuity of piecewise function: $f(x) = x^2$ for $x \neq 1$, $f(1) = 1$.

Answer: Continuous at $x = 1$. $\text{lim}_{x \to 1} x^2 = 1$ equals $f(1) = 1$.

Flashcard 15: Check continuity of $f(x) = \frac{1}{x}$ at $x = 1$.

Answer: Continuous at $x = 1$. All conditions satisfied: defined, limit exists, and they're equal.

Flashcard 16: Given $f(x) = x^2$ for $x \neq 1$ and $f(1) = 3$, determine continuity at $x = 1$.

Answer: Not continuous at $x = 1$. Limit is $1$ but function value is $3$.

Flashcard 17: Given $f(x) = |x|$, determine continuity at $x = 0$.

Answer: $f(x)$ is continuous at $x = 0$. Left limit equals right limit equals $f(0) = 0$.

Flashcard 18: Find if $f(x) = \frac{x^2 - 4}{x - 2}$ is continuous at $x = 2$.

Answer: Removable discontinuity at $x = 2$. Factor gives $\lim_{x \to 2} (x+2) = 4$ but $f(2)$ undefined.

Flashcard 19: State the definition of an infinite discontinuity.

Answer: A point where $f(x)$ approaches $\text{+}\text{ or }\text{-}\text{ infinity}$ as $x \to c$. The function grows without bound near that point.

Flashcard 20: Which condition fails if $f(x)$ is not defined at $x = c$?

Answer: The condition that $f(c)$ is defined. The function has no value at that point.

Flashcard 21: Find the type of discontinuity for $f(x) = \frac{1}{x-1}$ at $x = 1$.

Answer: Infinite discontinuity at $x = 1$. Vertical asymptote creates infinite discontinuity.

Flashcard 22: Determine if $f(x) = x^2 + 2x + 1$ is continuous for all $x$.

Answer: $f(x)$ is continuous for all $x$. Polynomial functions are continuous everywhere.

Flashcard 23: Determine continuity of $f(x) = \text{e}^x$ for all $x$.

Answer: $f(x)$ is continuous for all $x$. Exponential functions are continuous everywhere.

Flashcard 24: Identify the limit that must exist for continuity at $x = c$.

Answer: $\lim_{x \to c} f(x)$ must exist. Both one-sided limits must exist and be equal.

Flashcard 25: Identify the type of discontinuity: $f(x)$ undefined but limit exists.

Answer: Removable discontinuity. Can be fixed by defining the function at that point.

Flashcard 26: True or False: A function can be continuous at a point and not differentiable.

Answer: True. Functions can be continuous but have corners (like $|x|$ at $x=0$).

Flashcard 27: Determine continuity of $f(x) = \text{tan}(x)$ for $x \neq \frac{\text{n}\text{π}}{\text{2}}$.

Answer: $f(x)$ is continuous for $x \neq \frac{\text{n}\text{π}}{\text{2}}$. Tangent has vertical asymptotes at odd multiples of $\frac{\pi}{2}$.

Flashcard 28: What is the definition of continuity at a point $x = c$?

Answer: A function $f(x)$ is continuous at $x = c$ if $\text{lim}_{x \to c} f(x) = f(c)$. This states that the limit equals the function value at that point.

Flashcard 29: Identify the type of discontinuity if $\text{lim}_{x \to c} f(x) = \text{+}\text{ or }\text{-} \text{ infinity}$.

Answer: Infinite discontinuity. The function approaches infinity at that point.

Flashcard 30: Is the function $f(x) = [x]$ (greatest integer function) continuous for integer $x$?

Answer: No, it is not continuous at integer $x$. Floor function has jump discontinuities at every integer.