Continuity

For to be continuous for all , the left-hand limit at must equal the right-hand limit at . 1. For , . The left-hand limit as is: 2. For , . The right-hand limit as is: 3. For continuity at , these two limits must be equal: Thus, and are the values of that make continuous.

For the derivative to be continuous, the left-hand and right-hand derivatives must match at . This means we need to calculate the derivative of the function on both sides and ensure they match at . First, calculate the derivatives of each part of the function: 1. For , the derivative is: 2. For , the derivative is: At , the derivatives must match: Simplifying: Thus, . Substituting into the equation: Solving for : Thus, the values of and are and .

To ensure the function is continuous on , the values at the points where the function changes must be the same.
Step 1: Continuity at
At , the first and second parts of the function must be equal: This simplifies to: Thus, the first equation is:
Step 2: Continuity at
At , the second and third parts of the function must be equal: This simplifies to:
Step 3: Solve the system of equations
We now solve the system of equations: From Equation 2, . Substituting into Equation 1: Substituting into : Thus, the values of and are and .