Continuity And Differentiability

Possible graph of is as shown. There are to sharp turn, Hence cannot be differentiable at two point

Step 1: Analyze the continuity of on .
The function is continuous on as it is a composition of continuous functions.
Therefore, is also continuous on . Statement (B) is correct.

Step 2: Analyze the differentiability of on .
The derivative of is . This derivative is undefined at , which is in the interval . Thus, is not differentiable in . Statement (A) is correct.
Step 3: Check if .


Since , statement (C) is correct.

Step 4: Check if Rolle's theorem is applicable on .
Rolle's theorem requires three conditions to be met:
1. is continuous on . (Met) 2. is differentiable on . (Not met, as is not differentiable at )
3. . (Met)
Since the second condition is not satisfied, Rolle's theorem is not applicable on . Statement (D) is incorrect.

Step 1: Understand the function .
The absolute value function is defined as: Applying this to :
Step 2: Check for continuity at .
For a function to be continuous at , we need .
At , . Left-hand limit: Right-hand limit: Since , the function is continuous at .
Step 3: Check for differentiability at .
For a function to be differentiable at , the left-hand derivative must be equal to the right-hand derivative at that point.
We need to find for and .
For , , so .
For , , so .
Left-hand derivative at : Right-hand derivative at : Since and , the left-hand derivative is not equal to the right-hand derivative at . Therefore, is not differentiable at .