This question deals with a fundamental concept in real analysis concerning the relationship between continuity and uniform continuity on a specific type of domain.
Step 1: Understanding the Question:
The question asks whether the property of continuity for a function \(f(x)\) on a closed and bounded interval \( [a,b] \) is sufficient to guarantee the stronger property of uniform continuity on that same interval.
Step 2: Key Formula or Approach:
The answer to this question is a direct consequence of a major theorem in analysis, the Heine-Cantor Theorem.
Step 3: Detailed Explanation:
Continuity at a point means that for any desired closeness \(\epsilon\), we can find a neighborhood \(\delta\) around that point where the function values are within \(\epsilon\). The choice of \(\delta\) can depend on the point.
Uniform continuity on an interval means that for any \(\epsilon\), we can find a single \(\delta\) that works for the entire interval. This is a stronger condition.
The Heine-Cantor Theorem states that if a function is continuous on a compact set, then it is uniformly continuous on that set. In the context of real numbers (\(\mathbb{R}\)), a "compact set" is any set that is both closed and bounded.
The interval \( [a,b] \) given in the question is a closed and bounded interval, and therefore it is a compact set.
Since the function \(f(x)\) is continuous on the compact set \( [a,b] \), the Heine-Cantor Theorem directly implies that \(f(x)\) must also be uniformly continuous on \( [a,b] \).
Step 4: Final Answer:
Yes, a function continuous on a closed interval is always uniformly continuous on that interval.