Question

A continuous function on a closed and bounded interval is always:

• A. Differentiable
• B. Monotonic
• C. Bounded and attains its bounds
• D. Periodic

Hint:

Always associate “continuous + closed and bounded interval” with the Extreme Value Theorem.

Answer 1

The Correct Option is C

Step 1: Understanding the given condition.
The function is stated to be continuous on a closed and bounded interval. Let the interval be $[a, b]$, where $a$ and $b$ are real numbers, $a<b$, and both endpoints are included.
Step 2: Applying the Extreme Value Theorem.
According to the Extreme Value Theorem, if a function is continuous on a closed and bounded interval $[a, b]$, then:
• The function is bounded on $[a, b]$, and
• The function attains both its maximum and minimum values at least once in $[a, b]$.
Step 3: Analysis of the given options.
(A) Differentiable: Incorrect — continuity does not guarantee differentiability. A function can be continuous but not differentiable.
(B) Monotonic: Incorrect — a continuous function may increase and decrease within the interval.
(C) Bounded and attains its bounds: Correct — this follows directly from the Extreme Value Theorem.
(D) Periodic: Incorrect — periodicity is unrelated to continuity on a closed interval.
Step 4: Conclusion.
Since every continuous function on a closed and bounded interval is bounded and achieves both its maximum and minimum values, the correct answer is (C).