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: Which type of discontinuity occurs when $\lim_{x \to a} f(x) \neq f(a)$?

Answer: Removable discontinuity. The discontinuity can be 'removed' by redefining the function value.

Flashcard 2: 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 3: 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 4: 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 5: 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.

Flashcard 6: Is $f(x) = \text{sgn}(x)$ continuous at $x = 0$?

Answer: No, jump discontinuity at $x = 0$. Left and right limits are -1 and 1 respectively, creating a jump.

Flashcard 7: 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 8: 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 9: 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 10: 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 11: 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 12: 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 13: 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 14: 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 15: 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 16: 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 17: 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 18: 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 19: 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 20: 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 21: 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 22: 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 23: 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 24: 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 25: 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 26: Does $f(x) = \frac{x^2 - 4}{x - 2}$ have a discontinuity at $x = 2$?

Answer: Yes, removable discontinuity at $x = 2$. Function is undefined at $x=2$ where the denominator becomes zero.

Flashcard 27: Does $f(x) = \frac{x^2 - 1}{x - 1}$ have a discontinuity at $x = 1$?

Answer: Yes, removable discontinuity at $x = 1$. Function is undefined at $x=1$ where denominator equals zero.

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 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 30: For $f(x) = \text{sgn}(x)$, is $f(x)$ continuous at $x = 0$?

Answer: No, jump discontinuity at $x = 0$. The sign function jumps from -1 to 1 at $x=0$ with $f(0)=0$.

Flashcard 31: 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 32: For $f(x) = \frac{1}{x}$, is $f(x)$ continuous at $x = 0$?

Answer: No, non-removable discontinuity at $x = 0$. Function is undefined at $x=0$ and the limit approaches infinity.

Flashcard 33: 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 34: 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 35: 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 36: 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 37: 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 38: 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 39: Which type of discontinuity occurs when $\text{lim}_{x \to a} f(x) \neq f(a)$?

Answer: Removable discontinuity. The discontinuity can be 'removed' by redefining the function value.

Flashcard 40: 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 41: Does $f(x) = \frac{x^2 - 4}{x - 2}$ have a discontinuity at $x = 2$?

Answer: Yes, removable discontinuity at $x = 2$. Function is undefined at $x=2$ where the denominator becomes zero.

Flashcard 42: Is $f(x) = \text{sgn}(x)$ continuous at $x = 0$?

Answer: No, jump discontinuity at $x = 0$. Left and right limits are -1 and 1 respectively, creating a jump.

Flashcard 43: 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 44: 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 45: 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 46: 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 47: Identify the requirement for $f(a)$ for continuity at $x = a$.

Answer: $f(a)$ must be defined. The function must have a value at the point to be continuous there.

Flashcard 48: For $f(x) = \frac{1}{x}$, is $f(x)$ continuous at $x = 0$?

Answer: No, non-removable discontinuity at $x = 0$. Function is undefined at $x=0$ and the limit approaches infinity.

Flashcard 49: 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 50: 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 51: 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 52: For $f(x) = \text{sgn}(x)$, is $f(x)$ continuous at $x = 0$?

Answer: No, jump discontinuity at $x = 0$. The sign function jumps from -1 to 1 at $x=0$ with $f(0)=0$.

Flashcard 53: 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 54: 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 55: 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 56: 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 57: 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 $f(0) = |0| = 0$.