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: Interpreting the given information. 
We are told that a function is continuous on an interval that is both closed and bounded. Such an interval can be written as \([a, b]\), where both endpoints are included and \(a < b\).

Step 2: Relevant mathematical result.
A key theorem applicable here is the Extreme Value Theorem. It states that if a function is continuous on a closed and bounded interval, then:

• The function does not grow without limit (it is bounded).
• The function actually reaches a highest value (maximum) and a lowest value (minimum) somewhere in the interval.

Step 3: Evaluating the statements.
(A) Differentiable: Not necessarily true. Continuity alone does not ensure differentiability.
(B) Monotonic: Not guaranteed. A continuous function may increase in some parts and decrease in others.
(C) Bounded and attains its bounds: This is exactly what the Extreme Value Theorem guarantees.
(D) Periodic: Continuity on a finite interval does not imply periodic behavior.

Step 4: Final conclusion.
Every continuous function defined on a closed and bounded interval is bounded and achieves both its maximum and minimum values.

\[ \boxed{\text{Bounded and attains its bounds}} \]