Defining Continuity at a Point - AP Calculus BC

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

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

2. What is a removable discontinuity?

   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.

3. What condition characterizes non-removable discontinuities?

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

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

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

5. Is $f(x) = \text{sgn}(x)$ continuous at $x = 0$?

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

6. For $f(x) = 2x + 3$, is $f(x)$ continuous for all $x$?

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

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

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

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

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

9. Is $f(x) = \frac{1}{x^2}$ continuous at $x = 0$?

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

10. Is $f(x) = e^x$ continuous everywhere?

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

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

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

12. What is required for one-sided limits to exist at $x = a$?

    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.

13. Does $f(x) = x^2$ have any discontinuities?

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

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

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

15. Define a continuous function.

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

16. State the continuity condition for piecewise functions.

    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.

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

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

18. What condition must the limit satisfy for continuity at $x = a$?

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

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

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

20. Define continuity on an interval.

    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.

21. For $f(x) = \frac{1}{x}$, is $f(x)$ continuous at $x = 0$?

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

22. What is required for a function to be continuous from the left at $x = a$?

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

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

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

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

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

25. What is the definition of continuity at a point $x = a$?

    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.

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

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

27. Is $f(x) = \text{ln}(x)$ continuous for $x > 0$?

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

28. Which type of discontinuity occurs when $\text{lim}_{x \to a} f(x) \neq f(a)$?

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

29. What is the continuity requirement at $x = a$ for a piecewise function?

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

30. Does $f(x) = \frac{x^2 - 4}{x - 2}$ have a discontinuity at $x = 2$?

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

31. Define continuity at the endpoint of an interval.

    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.

32. Identify the requirement for $f(a)$ for continuity at $x = a$.

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

33. What is an infinite discontinuity?

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

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

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

35. Identify a non-removable discontinuity.

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

36. What is required for a function to be continuous from the right at $x = a$?

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

37. What is a jump discontinuity?

    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.

38. For $f(x) = |x|$, is $f(x)$ continuous at $x = 0$?

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

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