1.11 Defining Continuity at a Point

• What does the limit of a function $f(x)$ as $x$ approaches $a$ represent?


  The value $f(x)$ gets closer to as $x$ approaches $a$
• To calculate a limit, you always substitute the value $a$ directly into $f(x)$.


  False
• What are the three conditions for continuity of a function $f(x)$ at $x =$$a$?


  Function exists, limit exists, value equals limit
• Steps to check continuity of $f(x)$ at $x =$$2$


  1️⃣ $f(2)$ is defined

  2️⃣ $\lim_{x \to 2} f(x)$ exists

  3️⃣ $f(2) =$$\lim_{x \to 2} f(x)$
• Consider the function $f(x) =$$\frac{x^{2} - 4}{x - 2}$ for $x \neq 2$ and $f(2) =$$4$. Is $f(x)$ continuous at $x =$$2$?


  Yes
• For a function $f(x)$ to be continuous at $x =$$a$, $f(a)$ must be defined
• The limit of $f(x)$ as $x$ approaches $a$ can always be calculated by direct substitution.


  False
• What does it mean for a function $f(x)$ to be continuous at a point $x =$$a$?


  It satisfies three continuity conditions
• For continuity at $x =$$a$, $\lim_{x \to a} f(x)$ must exist
• The function $f(x) =$$\frac{x^{2} - 4}{x - 2}$ is continuous at $x =$$2$ if $f(2) =$$4$.
• What is the third condition for continuity at x = a</latex>?

  $f(a) =$$\lim_{x \to a} f(x)$
• For $\lim_{x \to a} f(x)$ to exist, both the left-hand limit and right-hand limit must be equal
• If a function $f(x)$ satisfies all three conditions for continuity at $x =$$a$, then it is continuous at that point.
• What are the three conditions for continuity of a function $f(x)$ at a point $x =$$a$?


  Function exists, limit exists, value equals limit
• For continuity at $x =$$a$, $\lim_{x \to a} f(x)$ must exist, meaning both the left-hand and right-hand limits must be equal
• A function $f(x)$ is continuous at a point $x =$$a$ if $f(a)$ is defined
• For a function to be continuous at $x =$$a$, the left-hand and right-hand limits must be equal.
• For a function to be continuous at $x =$$a$, $f(a)$ must equal \lim_{x \to a} f(x)</latex>
• Steps to check continuity of $f(x)$ at $x =$$2$ for $f(x) =$$\frac{x^{2} - 4}{x - 2}$ and $f(2) =$$4$


  1️⃣ $f(2) =$$4$ is defined

  2️⃣ $\lim_{x \to 2} \frac{x^{2} - 4}{x - 2} =$$4$ exists

  3️⃣ $f(2) =$$\lim_{x \to 2} f(x) =$$4$
• The function $f(x) =$$\frac{x^{2} - 4}{x - 2}$ is continuous at x =2</latex> if $f(2) =$$4$.
• A table summarizing continuity conditions includes checking if $\lim_{x \to 2} f(x)$ exists
• If a function is discontinuous at $x =$$a$, it must fail to meet all three continuity conditions.


  False
• A function is discontinuous at x = a</latex> if $f(a)$ is not defined
• Match the type of discontinuity with its definition:

  Removable ↔️ The limit exists but $f(a)$ is not defined or $f(a) \neq \lim_{x \to a} f(x)$

  Jump ↔️ The left-hand and right-hand limits exist but are not equal

  Infinite ↔️ The function approaches infinity or negative infinity as $x$ approaches $a$
• A removable discontinuity occurs when the limit exists but $f(a)$ is undefined.
• To calculate a limit, first substitute $a$ into $f(x)$. If you encounter an indeterminate form, simplify the function
• For a function to be continuous at $x =$$a$, its value at $x =$$a$ must equal the limit as $x$ approaches $a$.
• The condition "Function Exists" for continuity requires that $f(a)$ is defined.
• The condition "Limit Exists" for continuity requires that both the left-hand and right-hand limits are equal
• The condition "Value Equals Limit" for continuity requires that $f(a) =$$\lim_{x \to a} f(x)$.
• Consider the function $f(x) =$$\frac{x^{2} - 4}{x - 2}$ for $x \neq 2$ and $f(2) =$$4$. To check continuity at $x =$$2$, $f(2)$ is defined as 4
• What is the value of $\lim_{x \to 2} \frac{x^{2} - 4}{x - 2}$?


  4
• The function $f(x) =$$\frac{x^{2} - 4}{x - 2}$ is continuous at $x =$$2$.
• Match the type of discontinuity with its definition:

  Removable ↔️ The limit exists but $f(a)$ is not defined or $f(a) \neq \lim_{x \to a} f(x)$

  Jump ↔️ The left-hand and right-hand limits exist but are not equal

  Infinite ↔️ The function approaches infinity or negative infinity as $x$ approaches $a$
• Give an example of a removable discontinuity.

  $f(x) =$$\frac{x^{2} - 4}{x - 2}$ at $x =$$2$
• A jump discontinuity occurs when the left-hand and right-hand limits exist but are unequal
• What happens to the function in an infinite discontinuity as $x$ approaches $a$?


  It approaches infinity
• The function $f(x) =$ \begin{cases} x + 1 & x < 2 \\ 2x - 1 & x \geq 2 \end{cases} is continuous at $x =$$2$.
• The Intermediate Value Theorem (IVT) guarantees that a function takes on a specific value between two given points
• The Intermediate Value Theorem requires the function to be continuous on a closed interval.