Questions: Classify the statement as either true or false. If f is continuous at x=2, then f(2) does not exist.

Solution Steps

To determine if the statement is true or false, we need to understand the definition of continuity. A function \( f \) is continuous at \( x = 2 \) if the following three conditions are met:

1. \( f(2) \) is defined.
2. The limit of \( f(x) \) as \( x \) approaches 2 exists.
3. The limit of \( f(x) \) as \( x \) approaches 2 is equal to \( f(2) \).

Given that \( f \) is continuous at \( x = 2 \), it implies that \( f(2) \) must exist. Therefore, the statement "If \( f \) is continuous at \( x=2 \), then \( f(2) \) does not exist" is false.

To determine if the statement is true or false, we need to understand the definition of continuity. A function \( f \) is continuous at \( x = 2 \) if the following three conditions are met:

1. \( f(2) \) is defined.
2. \(\lim_{x \to 2} f(x)\) exists.
3. \(\lim_{x \to 2} f(x) = f(2)\).

The statement says: "If \( f \) is continuous at \( x=2 \), then \( f(2) \) does not exist."

Given that \( f \) is continuous at \( x = 2 \), it implies that \( f(2) \) must exist. This is because one of the conditions for continuity at \( x = 2 \) is that \( f(2) \) is defined.

Final Answer