Question

Which of the following statements is/are TRUE for the function \( f(x) \) shown in the figure given below? 

• A. \( f(x) \) is continuous at \( x = 0 \)
• B. \( f(x) \) is not continuous at \( x = 0 \)
• C. \( f(x) \) is differentiable at \( x = 0 \)
• D. \( f(x) \) is not differentiable at \( x = 0 \)

Hint:

To check if a function is continuous at a point, ensure that the function is defined at that point, and the left-hand and right-hand limits match.

Answer 1

The Correct Option is A, D

Step 1: Analyzing the Graph.
From the graph of \( f(x) \), the function is continuous at \( x = 0 \), as there is no discontinuity. The function passes through the origin with no breaks or jumps.
Step 2: Analyzing the Options.
- (A) Correct. The function is continuous at \( x = 0 \), as the left-hand and right-hand limits match and the function is defined at that point.
- (B) Incorrect. The function is continuous, as explained above.
- (C) Incorrect. While the function is continuous, it is not differentiable at \( x = 0 \) because the slope changes abruptly.
- (D) Incorrect. The function is continuous, but it is not differentiable at \( x = 0 \).
Step 3: Conclusion.
Thus, the correct answer is (A).