Continuity

To show that is continuous at , we need to verify that:

1. Step 1: Find :

2. Step 2: Find : We need to evaluate the limit of as . - For , . - For , . Thus, the limit of as from both sides is: Since both one-sided limits exist and are equal to , we have:

3. Step 3: Compare the limit with : Since , the function is continuous at .

Final Answer: Hence, is continuous at .

Step 1: Continuity of the function at is checked by verifying: Since both left-hand and right-hand limits exist and are equal to 0, the function is continuous at .

Step 2: Differentiability is checked by considering the derivative of . For , , and for , . At , the derivative does not exist because the left-hand derivative is not equal to the right-hand derivative. Thus, the function is not differentiable at .

Step 1: To determine continuity at , we check if the left-hand limit, right-hand limit, and the function value at are equal.

Step 2: The left-hand limit as is:

Step 3: The right-hand limit as is:

Step 4: Since the left-hand limit and the right-hand limit are not equal, the function is not continuous at .