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: 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 2: 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 3: 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 4: 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 5: 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 6: 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 7: 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 8: 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 9: 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 10: 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 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: 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 13: 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 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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 26: 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 27: 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 28: Determine the continuity of $f(x) = 5$ on its domain.

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

Flashcard 29: 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 30: Can a polynomial function have discontinuities?

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