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: Which condition fails if $\text{lim}_{x \to c} f(x) \neq f(c)$?

Answer: The condition that $\text{lim}_{x \to c} f(x) = f(c)$. The limit and function value don't match.

Flashcard 2: 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 3: 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 4: 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 5: 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 6: 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 7: 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 8: 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 9: 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 10: 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 11: 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 12: Which condition fails if $\text{lim}_{x \to c} f(x) \neq f(c)$?

Answer: The condition that $\text{lim}_{x \to c} f(x) = f(c)$. The limit and function value don't match.

Flashcard 13: Given $f(x) = \frac{1}{x}$, determine continuity at $x = 0$.

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

Flashcard 14: 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 15: 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 16: Find the discontinuity type: $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$.

Answer: Removable discontinuity. Factor cancels to give $\lim_{x \to 1} (x+1) = 2$ but $f(1)$ undefined.

Flashcard 17: 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 18: 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 19: 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 20: 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 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 continuity of $f(x) = \text{ln}(x)$ for $x > 0$.

Answer: $f(x)$ is continuous for $x > 0$. Natural logarithm is continuous on its domain.

Flashcard 23: 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 24: 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 25: 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 26: Given $f(x) = \frac{1}{x}$, determine continuity at $x = 0$.

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

Flashcard 27: 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 28: 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 29: Determine continuity of $f(x) = \text{ln}(x)$ for $x > 0$.

Answer: $f(x)$ is continuous for $x > 0$. Natural logarithm is continuous on its domain.

Flashcard 30: 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 31: 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 32: Identify the type of discontinuity where $\text{lim}_{x \to c^-} f(x) \neq \text{lim}_{x \to c^+} f(x)$.

Answer: Jump discontinuity. One-sided limits exist but differ in value.

Flashcard 33: 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 34: 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 35: What does $f(c)$ represent in the context of continuity?

Answer: The value of the function at the point $x = c$. This is the function's output at the specific point.

Flashcard 36: 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 37: 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 38: 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 39: Identify the type of discontinuity where $\text{lim}_{x \to c^-} f(x) \neq \text{lim}_{x \to c^+} f(x)$.

Answer: Jump discontinuity. One-sided limits exist but differ in value.

Flashcard 40: What is the second condition for continuity at a point $x = c$?

Answer: $\text{lim}_{x \to c} f(x)$ exists. Both left and right limits must exist and be equal.

Flashcard 41: 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 42: 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 43: 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.

Flashcard 44: 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 45: Determine if $f(x) = x^3$ is continuous for all $x$.

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

Flashcard 46: 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 47: What is the third condition for continuity at a point $x = c$?

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

Flashcard 48: 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 49: 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 50: 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 51: 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 52: Determine if $f(x) = x^3$ is continuous for all $x$.

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

Flashcard 53: What does $f(c)$ represent in the context of continuity?

Answer: The value of the function at the point $x = c$. This is the function's output at the specific point.

Flashcard 54: 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 55: 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 56: 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 57: 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 58: 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 59: 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 60: 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 61: 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.

Flashcard 62: 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 63: 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 64: 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 65: Find the discontinuity type: $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$.

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

Flashcard 66: 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 67: Find if $f(x) = \frac{x^2 - 4}{x - 2}$ is continuous at $x = 2$.

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

Flashcard 68: What is the second condition for continuity at a point $x = c$?

Answer: $\lim_{x \to c} f(x)$ exists. Both left and right limits must exist and be equal.

Flashcard 69: 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 70: 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 71: 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 72: 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 73: 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 74: 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 75: 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.