A function f(x) is continuous at x=a if f(a) is defined
For a function to be continuous at x=a, limx→af(x) must exist.
A function is continuous at x=a if \lim_{x \to a} f(x) = f(a)</latex>, meaning the limit equals the function's value
Match the condition with its description:
Function defined ↔️ f(a) must be a real number
Limit exists ↔️ limx→af(x) must approach a single value from both sides
Limit equals function value ↔️ limx→af(x)=f(a)
The function f(x) = \frac{x^{2} - 4}{x - 2}</latex> is defined at x=2.
False
The limit limx→2x−2x2−4 equals 4
The function f(x)=x−2x2−4 is continuous at x=2.
False
The function f(x)=x−2x2−4 is undefined atx = 2</latex> because it results in 0/0
The function f(x)=x−21 is undefined at x=2 because it results in division by zero
The limit limx→2x−21 exists.
False
The function f(x)=x2−3x+2 is continuous at x=1 because \lim_{x \to 1} f(x) = f(1) = 0</latex>, satisfying all three conditions for continuity
The function f(x)=x2−3x+2 is continuous at x=1.
The function f(x)= \begin{cases} x^{2} & \text{if } x \neq 1 \\ 2 & \text{if } x = 1 \end{cases} is discontinuous at x=1 because limx→1f(x)=f(1), failing the third condition for continuity
The piecewise function f(x) = \begin{cases} x^{2} & \text{if } x \neq 1 \\ 2 & \text{if } x = 1 \end{cases}</latex> is continuous at x=1.
False
What is the condition for continuity that requires limx→af(x)=f(a)?
Limit equals function value
The limit of f(x)=x2 as x→1 is 1
The function f(x) = \begin{cases} x^{2} & \text{if } x \neq 1 \\ 2 & \text{if } x = 1 \end{cases}</latex> is continuous at x=1.
False
Match the continuity condition with its description:
Function defined ↔️ f(a) must be a real number
Limit exists ↔️ limx→af(x) must exist
Limit equals function value ↔️ limx→af(x)=f(a)
For a function to be continuous at x=a, the limit limx→af(x) must exist
The condition limx→af(x)=f(a) ensures that the function value and the limit at x=a are equal.
Conditions for a function to be continuous from the right at x=a
1️⃣ f(a) is defined
2️⃣ limx→a+f(x) exists
3️⃣ limx→a+f(x)=f(a)
What type of continuity is checked at the right endpoint of an interval?
Left continuity
A function is continuous from the right at x=a if \lim_{x \to a^ + } f(x)</latex> equals f(a)