We will first prove that every differentiable function is continuous, and then we will examine the continuity and differentiability of at . 1. Step 1: Proving that every differentiable function is continuous If a function is differentiable at , it means that the derivative exists, which implies that: This implies that is continuous at , because: Thus, every differentiable function is continuous. 2. Step 2: Checking continuity of at To check the continuity of at , we must check if: First, find : Now, check the one-sided limits: - For , , so: - For , , so: Since both one-sided limits are equal to , we conclude that: Thus, is continuous at . 3. Step 3: Checking differentiability of at To check the differentiability at , we need to check if the left-hand and right-hand derivatives exist and are equal at . - For , , so .
- For , , so . Since the left-hand and right-hand derivatives are not equal, is not differentiable at . Final Answer: - The function is continuous at .
- The function is not differentiable at .