AP Calculus BC 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 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 the continuity requirement at $x = a$ for a piecewise function?

Answer: Limits from each piece must equal $f(a)$. Both pieces must approach the same value at the boundary point.

Flashcard 2: Define continuity at the endpoint of an interval.

Answer: A function is continuous at an endpoint if $\text{lim}$ from the interior equals the endpoint value. Only the one-sided limit from inside the interval needs to match.

Flashcard 3: Is $f(x) = \text{ln}(x)$ continuous for $x > 0$?

Answer: Yes, $f(x) = \text{ln}(x)$ is continuous for $x > 0$. Logarithm is defined and smooth for all positive real numbers.

Flashcard 4: Define a continuous function.

Answer: A function without any discontinuities over its entire domain. No breaks, jumps, or holes exist anywhere in the domain.

Flashcard 5: State the continuity condition for piecewise functions.

Answer: The limits from each piece must equal the function's value at the boundary. Left and right limits at transition points must equal the function value.

Flashcard 6: What is required for a function to be continuous from the right at $x = a$?

Answer: $\text{lim}_{x \to a^+} f(x) = f(a)$. The right-hand limit must equal the function value at the point.

Flashcard 7: For $f(x) = |x|$, is $f(x)$ continuous at $x = 0$?

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

Flashcard 8: Does $f(x) = x^2$ have any discontinuities?

Answer: No, $f(x) = x^2$ is continuous everywhere. Polynomial functions are continuous at every point in their domain.

Flashcard 9: What condition characterizes non-removable discontinuities?

Answer: $\text{lim}_{x \to a} f(x)$ does not exist. When the limit fails to exist, the discontinuity cannot be removed.

Flashcard 10: Evaluate $\text{lim}_{x \to 1} \frac{x^2 - 1}{x - 1}$. Is it continuous?

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

Flashcard 11: If $f(x) = \frac{x^2 - 1}{x - 1}$, is $f(x)$ continuous at $x = 1$?

Answer: No, removable discontinuity at $x = 1$. Function is undefined at $x=1$ where the denominator equals zero.

Flashcard 12: Find $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2}$. Is it continuous?

Answer: $\text{lim}_{x \to 2} \frac{x^2 - 4}{x - 2} = 4$. Removable discontinuity at $x = 2$. Factor and cancel to get $\lim_{x \to 2} (x+2) = 4$; undefined at $x=2$.

Flashcard 13: Which type of discontinuity occurs if $\text{lim}_{x \to a} f(x)$ does not exist?

Answer: Non-removable discontinuity. The discontinuity cannot be fixed by redefining a single point.

Flashcard 14: What is a removable discontinuity?

Answer: A discontinuity where $\text{lim}_{x \to a} f(x)$ exists but $\text{lim}_{x \to a} f(x) \neq f(a)$. The gap can be filled by redefining the function at that point.

Flashcard 15: What is a jump discontinuity?

Answer: A discontinuity where $\text{lim}_{x \to a^+} f(x) \neq \text{lim}_{x \to a^-} f(x)$. The function has different left and right limit values at the point.

Flashcard 16: Is $f(x) = e^x$ continuous everywhere?

Answer: Yes, $f(x) = e^x$ is continuous everywhere. Exponential functions are continuous throughout their entire domain.

Flashcard 17: Evaluate $\text{lim}_{x \to 0} \frac{\text{sin}x}{x}$. Is it continuous?

Answer: $\text{lim}_{x \to 0} \frac{\text{sin}x}{x} = 1$. Continuous at $x = 0$. This is a standard limit; $\frac{\sin x}{x}$ approaches 1 as $x \to 0$.

Flashcard 18: 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 is the formal definition combining limit existence and function value equality.

Flashcard 19: What is required for one-sided limits to exist at $x = a$?

Answer: Both $\text{lim}_{x \to a^+} f(x)$ and $\text{lim}_{x \to a^-} f(x)$ must exist. Each directional approach must have a finite limit value.

Flashcard 20: For $f(x) = 2x + 3$, is $f(x)$ continuous for all $x$?

Answer: Yes, $f(x) = 2x + 3$ is continuous for all $x$. Linear functions are continuous everywhere in the real numbers.

Flashcard 21: What condition must the limit satisfy for continuity at $x = a$?

Answer: $\text{lim}_{x \to a^+} f(x) = \text{lim}_{x \to a^-} f(x) = f(a)$. Both one-sided limits must exist and equal the function value.

Flashcard 22: Identify the type of discontinuity: $\text{lim}_{x \to 3} \frac{x^2 - 9}{x - 3} = 6$.

Answer: Removable discontinuity at $x = 3$. The limit exists but the function can be redefined to remove the gap.

Flashcard 23: Identify a non-removable discontinuity.

Answer: A discontinuity where $\text{lim}_{x \to a} f(x)$ does not exist. The limit failure creates an unfixable discontinuity.

Flashcard 24: Evaluate $\text{lim}_{x \to 0} \frac{1}{x}$. Is it continuous?

Answer: The limit does not exist. Non-removable discontinuity at $x = 0$. The function approaches infinity, so the limit doesn't exist.

Flashcard 25: Is $f(x) = \frac{1}{x^2}$ continuous at $x = 0$?

Answer: No, infinite discontinuity at $x = 0$. Function approaches positive infinity at $x=0$, creating infinite discontinuity.

Flashcard 26: Define continuity on an interval.

Answer: A function is continuous on an interval if it is continuous at every point in the interval. Every point in the interval must satisfy the continuity definition.

Flashcard 27: What is $\text{lim}_{x \to a} f(x)$ for continuity at $x = a$?

Answer: The limit must exist and be equal to $f(a)$. The limit and function value must be identical for continuity.

Flashcard 28: What is an infinite discontinuity?

Answer: A discontinuity where $f(x)$ approaches $\text{infinity}$ as $x$ approaches $a$. The function grows without bound as it approaches the point.

Flashcard 29: What is required for a function to be continuous from the left at $x = a$?

Answer: $\text{lim}_{x \to a^-} f(x) = f(a)$. The left-hand limit must equal the function value at the point.

Flashcard 30: Determine the type of discontinuity: $f(x) = \frac{1}{x}$ at $x = 0$.

Answer: Non-removable (infinite) discontinuity at $x = 0$. Function approaches infinity at $x=0$, creating an unbounded discontinuity.