Defining Continuity at a Point - AP Calculus AB

1. Which condition fails if $\text{lim}_{x \to c} f(x) \neq f(c)$?

   The condition that $\text{lim}_{x \to c} f(x) = f(c)$. The limit and function value don't match.

2. Find if $f(x) = \frac{x^2 - 4}{x - 2}$ is continuous at $x = 2$.

   Removable discontinuity at $x = 2$. Factor gives $\lim_{x \to 2} (x+2) = 4$ but $f(2)$ undefined.

3. What is the conclusion if $\text{lim}_{x \to c} f(x) = f(c)$ and $f(c)$ is defined?

   The function $f(x)$ is continuous at $x = c$. All three conditions for continuity are satisfied.

4. Check continuity of $f(x) = \frac{1}{x}$ at $x = 1$.

   Continuous at $x = 1$. All conditions satisfied: defined, limit exists, and they're equal.

5. State the definition of a jump discontinuity.

   A point where left and right limits exist but are not equal. The function jumps from one value to another at that point.

6. Identify the type of discontinuity if $\text{lim}_{x \to c} f(x) = \text{+}\text{ or }\text{-} \text{ infinity}$.

   Infinite discontinuity. The function approaches infinity at that point.

7. What is the third condition for continuity at a point $x = c$?

   $\lim_{x \to c} f(x) = f(c)$. The limit value must equal the function value.

8. Given $f(x) = \frac{1}{x}$, determine continuity at $x = 0$.

   $f(x)$ is not continuous at $x = 0$. Function undefined at $x = 0$ (division by zero).

9. Determine continuity of $f(x) = \text{tan}(x)$ for $x \neq \frac{\text{n}\text{π}}{\text{2}}$.

   $f(x)$ is continuous for $x \neq \frac{\text{n}\text{π}}{\text{2}}$. Tangent has vertical asymptotes at odd multiples of $\frac{\pi}{2}$.

10. Verify continuity of $f(x) = \frac{1}{x^2}$ at $x = 0$.

    $f(x)$ is not continuous at $x = 0$. Function undefined at $x = 0$ (division by zero).

11. Determine if $f(x) = \frac{1}{x}$ is continuous for $x \neq 0$.

    $f(x)$ is continuous for $x \neq 0$. Rational functions are continuous except where denominator is zero.

12. State whether $f(x) = \frac{\text{sin}(x)}{x}$ is continuous at $x = 0$.

    Removable discontinuity at $x = 0$. $\text{lim}_{x \to 0} \frac{\text{sin}(x)}{x} = 1$ but $f(0)$ undefined.

13. Identify the limit that must exist for continuity at $x = c$.

    $\lim_{x \to c} f(x)$ must exist. Both one-sided limits must exist and be equal.

14. Determine continuity of $f(x) = \text{ln}(x)$ for $x > 0$.

    $f(x)$ is continuous for $x > 0$. Natural logarithm is continuous on its domain.

15. Given $f(x) = x^2$ for $x \neq 1$ and $f(1) = 3$, determine continuity at $x = 1$.

    Not continuous at $x = 1$. Limit is $1$ but function value is $3$.

16. Identify the type of discontinuity where $\text{lim}_{x \to c^-} f(x) \neq \text{lim}_{x \to c^+} f(x)$.

    Jump discontinuity. One-sided limits exist but differ in value.

17. Which condition fails if $f(x)$ is not defined at $x = c$?

    The condition that $f(c)$ is defined. The function has no value at that point.

18. State the definition of an infinite discontinuity.

    A point where $f(x)$ approaches $\text{+}\text{ or }\text{-}\text{ infinity}$ as $x \to c$. The function grows without bound near that point.

19. What is the second condition for continuity at a point $x = c$?

    $\text{lim}_{x \to c} f(x)$ exists. Both left and right limits must exist and be equal.

20. Which condition fails if $\text{lim}_{x \to c} f(x)$ does not exist?

    The condition that $\text{lim}_{x \to c} f(x)$ exists. The left and right limits don't agree.

21. Determine continuity of $f(x) = \text{e}^x$ for all $x$.

    $f(x)$ is continuous for all $x$. Exponential functions are continuous everywhere.

22. Is the function $f(x) = [x]$ (greatest integer function) continuous for integer $x$?

    No, it is not continuous at integer $x$. Floor function has jump discontinuities at every integer.

23. Evaluate continuity of piecewise function: $f(x) = x^2$ for $x \neq 1$, $f(1) = 1$.

    Continuous at $x = 1$. $\text{lim}_{x \to 1} x^2 = 1$ equals $f(1) = 1$.

24. True or False: A function can be continuous at a point and not differentiable.

    True. Functions can be continuous but have corners (like $|x|$ at $x=0$).

25. Determine if $f(x) = x^3$ is continuous for all $x$.

    $f(x)$ is continuous for all $x$. Polynomial functions are continuous everywhere.

26. What does $f(c)$ represent in the context of continuity?

    The value of the function at the point $x = c$. This is the function's output at the specific point.

27. Find the type of discontinuity for $f(x) = \frac{1}{x-1}$ at $x = 1$.

    Infinite discontinuity at $x = 1$. Vertical asymptote creates infinite discontinuity.

28. Determine if $f(x) = \text{sin}(x)$ is continuous for all $x$.

    $f(x)$ is continuous for all $x$. Trigonometric functions are continuous on their domains.

29. What is the first condition for continuity at a point $x = c$?

    $f(c)$ is defined. The function must have a value at point $c$.

30. Identify the type of discontinuity: $f(x)$ undefined but limit exists.

    Removable discontinuity. Can be fixed by defining the function at that point.

31. State the definition of a removable discontinuity.

    A point where $f(x)$ is not defined or $\text{lim}_{x \to c} f(x) \neq f(c)$, but can be redefined. The gap can be filled by redefining the function at that point.

32. What is a continuous function?

    A function that is continuous at every point in its domain. No breaks, holes, or jumps anywhere in the domain.

33. Find the discontinuity type: $f(x) = \frac{x^2 - 1}{x - 1}$ at $x = 1$.

    Removable discontinuity. Factor cancels to give $\text{lim}_{x \to 1} (x+1) = 2$ but $f(1)$ undefined.

34. Determine if $f(x) = x^2 + 2x + 1$ is continuous for all $x$.

    $f(x)$ is continuous for all $x$. Polynomial functions are continuous everywhere.

35. State the Intermediate Value Theorem (IVT) condition for continuity.

    If $f$ is continuous on $[a, b]$ and $N$ is between $f(a)$ and $f(b)$, then $\text{exists } c \text{ in } (a, b)$ with $f(c) = N$. Continuous functions take on all intermediate values.

36. Given $f(x) = |x|$, determine continuity at $x = 0$.

    $f(x)$ is continuous at $x = 0$. Left limit equals right limit equals $f(0) = 0$.

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

    A function $f(x)$ is continuous at $x = c$ if $\text{lim}_{x \to c} f(x) = f(c)$. This states that the limit equals the function value at that point.

38. Find if $f(x) = \frac{x - 1}{x^2 - 1}$ is continuous at $x = 1$.

    Removable discontinuity at $x = 1$. Factor gives $\text{lim}_{x \to 1} \frac{1}{x+1} = \frac{1}{2}$ but $f(1)$ undefined.

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