Continuity

Step 1: Apply continuity condition.
For continuity at and , the function should not have any breaks at these points. By solving for and , we find and .
Thus, the correct answer is (2).

Step 1: Check for continuity.
For the function to be continuous at , the limit as approaches must equal the function value at .
Step 2: Evaluate the limit.
By simplifying the expression for the limit, we find that . Final Answer:

Step 1: Left hand limit as : Hence .
Step 2: Right hand limit as :
Step 3: Continuity gives:

Step 1: f(x)=x³-1 is a polynomial, hence continuous everywhere, so statement I is true.
Step 2: Root of x³-1=0 is x=1, which is not in (-1,1). Thus statement II is also true? But root lies at boundary x=1, so statement II (“no root in (-1,1)”) is true. However, question asks correctness: only continuity is being emphasized as sure property.

Absolute value and polynomial functions are continuous everywhere. Hence f(x) is continuous for all x∈[-1,1].