Question

If \( A \) denotes the set of continuous functions and \( B \) denotes the set of differentiable functions, then which of the following depicts the correct relation between set \( A \) and \( B \)?

• A.

  

• B.

  

• C.

  

• D.

  

Hint:

In the context of functions, differentiable functions are always continuous, but continuous functions are not always differentiable.

Answer 1

The Correct Option is A

To solve the problem, we need to understand the relationship between the set of continuous functions (A) and differentiable functions (B).

1. Mathematical Insight:
- Every differentiable function is continuous.
- However, not every continuous function is differentiable (e.g., \( f(x) = |x| \) is continuous but not differentiable at \( x = 0 \)).

This means the set of differentiable functions (B) is a subset of the set of continuous functions (A).

2. Diagram Interpretation:
We are looking for a diagram where the set B (differentiable functions) is completely inside set A (continuous functions).

3. Evaluate the Options:
- Option (A): Set B is inside A — ✔️
- Option (B): Set A is inside B — ❌
- Option (C): A and B overlap partially — ❌
- Option (D): A and B are disjoint — ❌

Final Answer:
The correct option is (A), where the set of differentiable functions is completely contained within the set of continuous functions.