Question

Let \( D = \{(x, y) \in \mathbb{R}^2 : |x| + |y| \le 1\} \) and \( f : D \to \mathbb{R} \) be a non-constant continuous function. Which of the following is TRUE?
 

• A. The range of \( f \) is unbounded
• B. The range of \( f \) is a union of open intervals
• C. The range of \( f \) is a closed interval
• D. The range of \( f \) is a union of at least two disjoint closed intervals

Hint:

A continuous function on a compact (closed and bounded) domain always has a closed and bounded range, forming a closed interval in \( \mathbb{R}. \)

Answer 1

The Correct Option is C

Step 1: Analyze the domain.
The set \( D = \{(x, y): |x| + |y| \le 1\} \) is a closed and bounded region in \( \mathbb{R}^2 \) (a diamond-shaped area). Hence, \( D \) is compact.

Step 2: Property of continuous functions on compact sets.
If a function \( f \) is continuous on a compact set, then \( f \) attains its maximum and minimum values and its range is closed and bounded.

Step 3: Since \( f \) is non-constant,
its range will include more than one value but will still be a continuous closed interval between the minimum and maximum.

Final Answer: The range of \( f \) is a closed interval.