AP Calculus BC Flashcards: Confirming Continuity Over An Interval

What this deck covers

This deck focuses on Confirming Continuity Over An Interval, giving you a quick way to review the definitions, rules, and examples that matter most for AP Calculus BC.

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: What is $\text{lim}_{x \to 2} (x^2 - 4)/(x - 2)$?

Answer: The limit is $4$, indicating a removable discontinuity. Factor $(x-2)(x+2)$ and cancel to get $\lim_{x \to 2} (x+2) = 4$.

Flashcard 2: Is $f(x) = x^3$ continuous for all real numbers?

Answer: Yes, $f(x) = x^3$ is continuous everywhere. Cubic polynomials have no domain restrictions.

Flashcard 3: State the three conditions for continuity at a point.

Answer: $f(a)$ is defined, $\text{lim}_{x \to a} f(x)$ exists, $\text{lim}_{x \to a} f(x) = f(a)$. All three must hold for continuity to be confirmed.

Flashcard 4: Explain if $f(x) = x^{-2}$ is continuous at $x = 0$.

Answer: $f(x) = x^{-2}$ is not continuous at $x = 0$. The function is undefined due to division by zero.

Flashcard 5: What is a removable discontinuity?

Answer: A discontinuity that can be removed by redefining $f(a)$. The limit exists but doesn't equal the function value.

Flashcard 6: What is the definition of continuity at a point $x = a$?

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

Flashcard 7: What must be true for a function's one-sided limits at $x = a$ for continuity?

Answer: The one-sided limits must be equal at $x = a$. This ensures the limit exists at the point.

Flashcard 8: What is the continuity status of $f(x) = \text{ln}(x)$ at $x = 0$?

Answer: $f(x) = \text{ln}(x)$ is not continuous at $x = 0$. Natural log is undefined at zero and negative values.

Flashcard 9: Can a polynomial function have discontinuities?

Answer: No, polynomial functions are continuous everywhere. Polynomials have no restrictions on their domain.

Flashcard 10: Identify the type of discontinuity in $f(x) = \frac{1}{x^2 - 1}$ at $x = 1$.

Answer: Infinite discontinuity at $x = 1$. The function approaches infinity as $x$ approaches 1.

Flashcard 11: What is the Intermediate Value Theorem?

Answer: If $f(x)$ is continuous on $[a, b]$, $f(c)$ takes every value between $f(a)$ and $f(b)$. Guarantees existence of intermediate values on continuous functions.

Flashcard 12: What is the Intermediate Value Theorem?

Answer: If $f(x)$ is continuous on $[a, b]$, $f(c)$ takes every value between $f(a)$ and $f(b)$. Guarantees existence of intermediate values on continuous functions.

Flashcard 13: Determine continuity of $f(x) = \frac{1}{x}$ at $x = 0$.

Answer: $f(x)$ is not continuous at $x = 0$ as it is undefined. Division by zero makes the function undefined at this point.

Flashcard 14: Determine if $f(x) = 3x^2 - 2x + 1$ is continuous at $x = 2$.

Answer: $f(x)$ is continuous at $x = 2$ since it is a polynomial. Polynomials are continuous at every point in their domain.

Flashcard 15: Determine continuity of $f(x) = \frac{1}{x}$ at $x = 0$.

Answer: $f(x)$ is not continuous at $x = 0$ as it is undefined. Division by zero makes the function undefined at this point.

Flashcard 16: What is the role of $\text{lim}_{x \to a} f(x)$ in confirming continuity?

Answer: It must equal $f(a)$ for continuity at $x = a$. The limit must equal the function value for continuity.

Flashcard 17: For $f(x) = \frac{x^2-1}{x-1}$, identify the discontinuity at $x = 1$.

Answer: Removable discontinuity at $x = 1$. Factor and cancel to find $\lim_{x \to 1} (x+1) = 2$.

Flashcard 18: Can a rational function have discontinuities?

Answer: Yes, at points where the denominator is zero. Discontinuities occur where denominators equal zero.

Flashcard 19: Is $f(x) = \text{sin}(x)$ continuous at $x = \frac{\text{pi}}{2}$?

Answer: Yes, $f(x) = \text{sin}(x)$ is continuous at $x = \frac{\text{pi}}{2}$. Sine function is continuous at all points in its domain.

Flashcard 20: Is $f(x) = |x|$ continuous at $x = 0$?

Answer: Yes, $f(x) = |x|$ is continuous at $x = 0$. Both one-sided limits equal 0, matching $f(0) = 0$.

Flashcard 21: Determine the continuity of $f(x) = 5$ on its domain.

Answer: $f(x) = 5$ is continuous everywhere. Constant functions are continuous everywhere.

Flashcard 22: What is the role of $\text{lim}_{x \to a} f(x)$ in confirming continuity?

Answer: It must equal $f(a)$ for continuity at $x = a$. The limit must equal the function value for continuity.

Flashcard 23: What is the continuity status of $f(x) = \frac{1}{x}$ on $\text{R}$?

Answer: $f(x)$ is continuous on $\text{R} \backslash \text{0}$. Continuous everywhere except where the denominator is zero.

Flashcard 24: Explain the continuity of $f(x) = \text{sin}(x)$ on $\text{R}$.

Answer: $f(x) = \text{sin}(x)$ is continuous on all real numbers. Trigonometric sine function has no domain restrictions.

Flashcard 25: Is $f(x) = \frac{1}{x-2}$ continuous at $x = 2$?

Answer: No, $f(x)$ is not continuous at $x = 2$. The denominator equals zero, making the function undefined.

Flashcard 26: Define continuity for a function on a closed interval $[a, b]$.

Answer: $f(x)$ is continuous on $[a, b]$ if continuous on $(a, b)$ and limits match at $a$, $b$. Requires continuity at interior points and proper one-sided limits.

Flashcard 27: What is a removable discontinuity?

Answer: A discontinuity that can be removed by redefining $f(a)$. The limit exists but doesn't equal the function value.

Flashcard 28: Is $f(x) = |x|$ continuous at $x = 0$?

Answer: Yes, $f(x) = |x|$ is continuous at $x = 0$. Both one-sided limits equal 0, matching $f(0) = 0$.

Flashcard 29: Determine continuity of $f(x) = 2x + 3$ on $(-\text{infinity}, \text{infinity})$.

Answer: $f(x)$ is continuous for all real numbers. Linear functions are continuous everywhere.

Flashcard 30: Identify the type of discontinuity in $f(x) = \frac{1}{x^2 - 1}$ at $x = 1$.

Answer: Infinite discontinuity at $x = 1$. The function approaches infinity as $x$ approaches 1.

Flashcard 31: State the limit condition for continuity over an open interval $(a, b)$.

Answer: $f(x)$ is continuous if $\text{lim}_{x \to c} f(x) = f(c)$ for all $c \text{ in } (a, b)$. This is the fundamental definition of continuity over intervals.

Flashcard 32: What must be true for a function's one-sided limits at $x = a$ for continuity?

Answer: The one-sided limits must be equal at $x = a$. This ensures the limit exists at the point.

Flashcard 33: Explain the continuity of $f(x) = \text{sin}(x)$ on $\text{R}$.

Answer: $f(x) = \text{sin}(x)$ is continuous on all real numbers. Trigonometric sine function has no domain restrictions.

Flashcard 34: Determine continuity of $f(x) = 2x + 3$ on $(-\text{infinity}, \text{infinity})$.

Answer: $f(x)$ is continuous for all real numbers. Linear functions are continuous everywhere.

Flashcard 35: Determine if $f(x) = 3x^2 - 2x + 1$ is continuous at $x = 2$.

Answer: $f(x)$ is continuous at $x = 2$ since it is a polynomial. Polynomials are continuous at every point in their domain.

Flashcard 36: What does it mean for $\text{lim}_{x \to a} f(x)$ to exist?

Answer: Both $\text{lim}_{x \to a^-} f(x)$ and $\text{lim}_{x \to a^+} f(x)$ exist and are equal. The left and right limits must converge to the same value.

Flashcard 37: What is the continuity status of $f(x) = \text{ln}(x)$ at $x = 0$?

Answer: $f(x) = \text{ln}(x)$ is not continuous at $x = 0$. Natural log is undefined at zero and negative values.

Flashcard 38: What is the continuity status of $f(x) = \frac{1}{x}$ on $\text{R}$?

Answer: $f(x)$ is continuous on $\text{R} \backslash \text{0}$. Continuous everywhere except where the denominator is zero.

Flashcard 39: What is the definition of continuity at a point $x = a$?

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

Flashcard 40: Identify the type of discontinuity if $\text{lim}_{x \to a} f(x)$ does not exist.

Answer: There is an infinite or jump discontinuity at $x = a$. When one-sided limits differ or approach infinity.

Flashcard 41: Is $f(x) = \frac{1}{x-2}$ continuous at $x = 2$?

Answer: No, $f(x)$ is not continuous at $x = 2$. The denominator equals zero, making the function undefined.

Flashcard 42: Identify the type of discontinuity if $\text{lim}_{x \to a} f(x)$ does not exist.

Answer: There is an infinite or jump discontinuity at $x = a$. When one-sided limits differ or approach infinity.

Flashcard 43: What is the limit condition for $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$?

Answer: The limit is $2$, indicating a removable discontinuity. Factor and cancel: $\lim_{x \to 1} (x+1) = 2$.

Flashcard 44: What is the continuity status of $f(x) = \text{cos}(x)$?

Answer: $f(x) = \text{cos}(x)$ is continuous on all real numbers. Trigonometric cosine function has no domain restrictions.

Flashcard 45: What is the continuity status of $f(x) = \text{cos}(x)$?

Answer: $f(x) = \text{cos}(x)$ is continuous on all real numbers. Trigonometric cosine function has no domain restrictions.

Flashcard 46: Can a rational function have discontinuities?

Answer: Yes, at points where the denominator is zero. Discontinuities occur where denominators equal zero.

Flashcard 47: Determine the continuity of $f(x) = \frac{x}{x^2 - 1}$ at $x = 1$.

Answer: $f(x)$ is not continuous at $x = 1$ due to division by zero. The denominator $(x-1)(x+1)$ equals zero at $x = 1$.

Flashcard 48: What does it mean for $\text{lim}_{x \to a} f(x)$ to exist?

Answer: Both $\text{lim}_{x \to a^-} f(x)$ and $\text{lim}_{x \to a^+} f(x)$ exist and are equal. The left and right limits must converge to the same value.

Flashcard 49: For $f(x) = \frac{x^2-1}{x-1}$, identify the discontinuity at $x = 1$.

Answer: Removable discontinuity at $x = 1$. Factor and cancel to find $\lim_{x \to 1} (x+1) = 2$.

Flashcard 50: Is $f(x) = \text{sin}(x)$ continuous at $x = \frac{\text{pi}}{2}$?

Answer: Yes, $f(x) = \text{sin}(x)$ is continuous at $x = \frac{\text{pi}}{2}$. Sine function is continuous at all points in its domain.

Flashcard 51: Explain if $f(x) = x^{-2}$ is continuous at $x = 0$.

Answer: $f(x) = x^{-2}$ is not continuous at $x = 0$. The function is undefined due to division by zero.

Flashcard 52: What is $\text{lim}_{x \to 2} (x^2 - 4)/(x - 2)$?

Answer: The limit is $4$, indicating a removable discontinuity. Factor $(x-2)(x+2)$ and cancel to get $\lim_{x \to 2} (x+2) = 4$.

Flashcard 53: Is $f(x) = x^3$ continuous for all real numbers?

Answer: Yes, $f(x) = x^3$ is continuous everywhere. Cubic polynomials have no domain restrictions.

Flashcard 54: State the three conditions for continuity at a point.

Answer: $f(a)$ is defined, $\text{lim}_{x \to a} f(x)$ exists, $\text{lim}_{x \to a} f(x) = f(a)$. All three must hold for continuity to be confirmed.

Flashcard 55: Identify the type of discontinuity for a step function.

Answer: A step function has a jump discontinuity. Function values change abruptly at certain points.

Flashcard 56: What is necessary for a function to be continuous on a closed interval $[a, b]$?

Answer: The function must be continuous on $(a, b)$ and limits must match at $a$ and $b$. Combines interior continuity with proper boundary behavior.

Flashcard 57: What is necessary for a function to be continuous on a closed interval $[a, b]$?

Answer: The function must be continuous on $(a, b)$ and limits must match at $a$ and $b$. Combines interior continuity with proper boundary behavior.

Flashcard 58: Determine the continuity of $f(x) = 5$ on its domain.

Answer: $f(x) = 5$ is continuous everywhere. Constant functions are continuous everywhere.

Flashcard 59: Can a polynomial function have discontinuities?

Answer: No, polynomial functions are continuous everywhere. Polynomials have no restrictions on their domain.

Flashcard 60: State the limit condition for continuity over an open interval $(a, b)$.

Answer: $f(x)$ is continuous if $\text{lim}_{x \to c} f(x) = f(c)$ for all $c \text{ in } (a, b)$. This is the fundamental definition of continuity over intervals.

Flashcard 61: Determine the continuity of $f(x) = \frac{x}{x^2 - 1}$ at $x = 1$.

Answer: $f(x)$ is not continuous at $x = 1$ due to division by zero. The denominator $(x-1)(x+1)$ equals zero at $x = 1$.

Flashcard 62: Identify the type of discontinuity for a step function.

Answer: A step function has a jump discontinuity. Function values change abruptly at certain points.

Flashcard 63: What is the limit condition for $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$?

Answer: The limit is $2$, indicating a removable discontinuity. Factor and cancel: $\lim_{x \to 1} (x+1) = 2$.

Flashcard 64: Define continuity for a function on a closed interval $[a, b]$.

Answer: $f(x)$ is continuous on $[a, b]$ if continuous on $(a, b)$ and limits match at $a$, $b$. Requires continuity at interior points and proper one-sided limits.