1.11 Defining Continuity at a Point

• Match the condition with its description for continuity:

  f(c) is defined ↔️ The function value exists at c

  $\lim_{x \to c} f(x)$ exists ↔️ The limit as x approaches c exists

  $\lim_{x \to c} f(x) =$$f(c)$ ↔️ The limit equals the function value
• For a function to be continuous at x = a, the limit of f(x) as x approaches a must exist
• The concept of limits is crucial in determining if a function is continuous at a point.


  True
• Match the continuity condition with its discontinuity counterpart:

  Limit exists ↔️ Limit does not exist

  Value is defined ↔️ Value is undefined

  Limit = Value ↔️ Limit ≠ Value
• If the limit of f(x) as x approaches a does not exist, then f(x) is discontinuous at x = a.


  True
• The third condition for continuity at x = a requires that the limit of f(x) as x approaches a equals f(a).

  True
• Match the condition with its requirement for continuity or discontinuity:

  Limit of f(x) as x→a ↔️ Exists for continuity, does not exist for discontinuity

  Value of f(x) at x=a ↔️ Defined for continuity, undefined for discontinuity

  Limit = Value ↔️ Yes for continuity, no for discontinuity
• The right-hand limit is the limit of f(x) as x approaches c from larger values
• What is the value of f(x) at x = 0 for the function f(x) = \begin{cases} x^{2} & x \neq 0 \\ 1 & x = 0 \end{cases}</latex>?

  1
• For continuity, the limit of $f(x)$ as $x$ approaches a</latex> must exist
• Match the continuity and discontinuity conditions with their descriptions:

  Continuity: Limit Exists ↔️ $\lim_{x \to a} f(x)$ exists

  Continuity: Value Defined ↔️ $f(a)$ is defined

  Continuity: Limit = Value ↔️ $\lim_{x \to a} f(x) =$$f(a)$
• What is the value of \lim_{x \to 2^ - } f(x)</latex> for $f(x) =$ \begin{cases} x + 1 & x \le 2 \\ 3 & x > 2 \end{cases}?


  3
• The first condition for continuity at a point is that the function value at that point must be defined.

  True
• What is the value of $f(2)$ for f(x) = \begin{cases} x + 1 & x \leq 2 \\ 3 & x > 2 \end{cases}</latex>?


  3
• A function f(x) is continuous at a point c if f(c) is defined
• The conditions for continuity ensure that there is no break, jump, or hole in the graph of f(x) at x = c.
• Summarize the three conditions for continuity in the correct order:

  1️⃣ Limit exists

  2️⃣ Value is defined

  3️⃣ Limit equals value
• For a function f(x) to be continuous at x = a, the limit of f(x) as x approaches a must exist
• For a function f(x) to be continuous at x = c, the limit of f(x) as x approaches c must equal f(c)
• The second condition for continuity at x = a is that the value of f(x) at x = a must be defined
• What is the term for a function that does not satisfy all conditions for continuity at a point?

  Discontinuous
• What condition must be satisfied for a limit to exist at x = c?

  Left-hand limit = Right-hand limit
• The function $g(x) =$$\frac{1}{x}$ is discontinuous at x = 0 because division by zero is undefined.


  True
• A function is discontinuous at a point if its limit does not exist as x approaches that point.

  True
• A function is discontinuous if its limit as x approaches a is equal to $f(a)$.


  False
• The left-hand limit of $f(x)$ as $x$ approaches c</latex> is denoted as $\lim_{x \to c^ - } f(x)$
• Match the continuity conditions with their explanations:

  Continuity: Defined Value ↔️ Ensures no hole or gap in the graph

  Discontinuity: Undefined Value ↔️ Indicates a hole or vertical asymptote
• For the function $f(x) =$ \begin{cases} x + 1 & x \leq 2 \\ 3 & x > 2 \end{cases} at $x =$$2$, the limit exists because the left-hand and right-hand limits both equal 3
• The limit of f(x) as x approaches c must exist for f(x) to be continuous at c.


  True
• The value of f(x) at x = a must be defined for f(x) to be continuous at x = a.

  True
• The limit of f(x) as x approaches a must equal the value of f(x) at x = a
• A function f(x) is continuous at x = c if lim x→c f(x) exists.


  True
• What is the first condition for a function to be continuous at x = a?

  Limit exists
• The left-hand limit is the limit of f(x) as x approaches c from smaller values.

  True
• Match the limit type with its direction of approach and existence condition:

  Left-hand limit ↔️ From the left, must exist

  Right-hand limit ↔️ From the right, must exist

  Standard limit ↔️ From both sides, Left-hand limit = Right-hand limit
• For a function $f(x)$ to be continuous at $x =$$a$, the limit as $x$ approaches $a$ must equal the value of $f(a)$
• Match the continuity conditions with their definitions:

  $f(a)$ is defined ↔️ The function has a value at $x =$$a$

  $\lim_{x \to a} f(x)$ exists ↔️ The limit approaches the same value from both sides

  $\lim_{x \to a} f(x) =$$f(a)$ ↔️ The limit equals the function's value at $x =$$a$
• Arrange the three conditions for continuity in their logical order:

  1️⃣ Limit Exists: $\lim_{x \to a} f(x)$ exists

  2️⃣ Defined Value: $f(a)$ is defined

  3️⃣ Limit = Value: $\lim_{x \to a} f(x) =$$f(a)$
• A standard limit exists at $x =$$c$ if the left-hand and right-hand limits are equal.


  True
• For the function $f(x) =$ \begin{cases} x + 1 & x \le 2 \\ 3 & x > 2 \end{cases}, the right-hand limit as $x$ approaches 2 is 3