Continuity And Differentiability

The function is defined as: To prove that the function is not differentiable at , we need to check if the derivative from the left and right at are the same.

Step 1: Left-hand derivative. The left-hand derivative is the derivative of for . The derivative of is: So, the left-hand derivative at is .

Step 2: Right-hand derivative. The right-hand derivative is the derivative of for . The derivative of is: So, the right-hand derivative at is .

Step 3: Conclusion. Since the left-hand and right-hand derivatives at are not equal, the function is not differentiable at . Conclusion: Thus, is not differentiable at .

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 .