Question

If a function \( f(x) \) is continuous on a closed interval \( [a,b] \), is it necessarily uniformly continuous?

• A. Yes, always uniformly continuous
• B. No, never uniformly continuous
• C. Only if differentiable
• D. Only if bounded

Hint:

Continuous on closed interval \( \Rightarrow \) Uniformly continuous (Always true!)

Answer 1

The Correct Option is A

Concept: Heine–Cantor Theorem
A very important result in real analysis states:

• If a function is continuous on a closed and bounded interval \( [a,b] \),
• Then it is uniformly continuous on that interval.

Step 1: Understand continuity vs uniform continuity

• Continuity: For every point \(x\), we can find a \(\delta\) depending on \(x\).
• Uniform continuity: A single \(\delta\) works for all \(x \in [a,b]\).

Step 2: Apply the theorem
Since the function is continuous on a closed interval \( [a,b] \), it satisfies: \[ \text{Uniform continuity on } [a,b] \]
Step 3: Why closed interval matters
Closed intervals are:

• Bounded
• Contain all limit points

This ensures no “escape” of function behavior at endpoints. Conclusion: \[ \text{The function is always uniformly continuous.} \]