Question

Let \( f : \mathbb{R} \to \mathbb{R} \) be a function and let \( I \) be a bounded open interval in \( \mathbb{R} \). Define
\[ W(f, I) = \sup \{ f(x) | x \in I \} - \inf \{ f(x) | x \in I \} \] Which one of the following is FALSE?

• A. \( W(f, J_1) \leq W(f, J_2) \) if \( J_1 \subseteq J_2 \)
• B. If \( f \) is a bounded function in \( I \) and \( J_1 \supseteq J_2 \supseteq \cdots \supseteq J_n \) such that the length of the interval \( J_n \) tends to 0 as \( n \to \infty \), then \( \lim_{n \to \infty} W(f, J_n) = 0 \)
• C. If \( f \) is discontinuous at a point \( a \in I \), then \( W(f, I) \neq 0 \)
• D. If \( f \) is continuous at a point \( a \in I \), then for any given \( \epsilon > 0 \) there exists an interval \( I \subseteq J \) such that \( W(f, I) < \epsilon \)

Hint:

When working with functions, the supremum and infimum are influenced by the continuity of the function. Discontinuities do not always imply a non-zero difference between the supremum and infimum.

Answer 1

The Correct Option is B

Step 1: Analyzing statement (C).
If \( f \) is discontinuous at a point \( a \in I \), it does not necessarily imply that \( W(f, I) \neq 0 \). It is possible for the supremum and infimum to be equal, even if \( f \) is discontinuous at a point. Thus, statement (C) is FALSE.

Step 2: Checking the other options.
(A) is true because \( W(f, J_1) \leq W(f, J_2) \) for \( J_1 \subseteq J_2 \).
(B) is true because as the length of the interval \( J_n \) tends to 0, the difference between the supremum and infimum of \( f(x) \) tends to 0.
(D) is true because continuity implies that for any \( \epsilon > 0 \), there exists an interval where \( W(f, I) \) is less than \( \epsilon \).

Step 3: Conclusion.
The correct answer is (C) If \( f \) is discontinuous at a point \( a \in I \), then \( W(f, I) \neq 0 \).