Question

Let \( f: [0, \infty) \to \mathbb{R} \) be a function, where \( \mathbb{R} \) denotes the set of all real numbers. Then which one of the following statements is true?

• A. If \( f \) is bounded and continuous, then \( f \) is uniformly continuous
• B. If \( f \) is uniformly continuous, then \( \lim_{x \to \infty} f(x) \) exists
• C. If \( f \) is uniformly continuous, then the function \( g(x) = f(x) \sin x \) is also uniformly continuous
• D. If \( f \) is continuous and \( \lim_{x \to \infty} f(x) \) is finite, then \( f \) is uniformly continuous

Hint:

Uniform continuity can often be ensured by limiting the behavior of the function at infinity or by restricting the domain to a compact set.

Answer 1

The Correct Option is D

Let us analyze the given statements one by one: 
Step 1: Bounded and continuous implies uniform continuity. 
If \( f \) is bounded and continuous on \( [0, \infty) \), this does not guarantee uniform continuity. Uniform continuity requires the behavior of \( f \) to be controlled uniformly for all points in the domain, which is not guaranteed by just boundedness and continuity. 
Step 2: Uniform continuity does not imply a limit at infinity. 
The fact that \( f \) is uniformly continuous does not necessarily imply that \( \lim_{x \to \infty} f(x) \) exists. A function can be uniformly continuous without having a limit as \( x \to \infty \). 
Step 3: Uniform continuity does not guarantee uniform continuity of \( g(x) = f(x) \sin x \). 
While \( f \) is uniformly continuous, multiplying by \( \sin x \), which oscillates, can cause \( g(x) \) to fail to be uniformly continuous because the oscillations may disrupt the uniformity. 
Step 4: Continuity and a finite limit at infinity imply uniform continuity. 
If \( f \) is continuous on \( [0, \infty) \) and \( \lim_{x \to \infty} f(x) \) is finite, then \( f \) must be uniformly continuous because the behavior of \( f(x) \) becomes stable as \( x \) grows larger, ensuring the function remains controlled. 
Final Answer: \[ \boxed{\text{(D) If } f \text{ is continuous and } \lim_{x \to \infty} f(x) \text{ is finite, then } f \text{ is uniformly continuous}}. \]