Domain Of A Function

The function is considered. We need to determine its continuity and differentiability on the interval .

Step 1: Analyze Continuity

The absolute value function is continuous everywhere, meaning is also continuous everywhere, including on .

Conclusion on Continuity: The function is continuous on .

Step 2: Analyze Differentiability

The differentiability of requires us to examine the point where because absolute value functions may not be differentiable at points where the expression inside the absolute value changes sign.

Specifically, at , the left-hand derivative and right-hand derivative are calculated as follows:

• For , , thus .
• For , , thus .

At , the left-hand derivative is and the right-hand derivative is . Since these derivatives are not equal, is not differentiable at .

Conclusion on Differentiability: The function is not differentiable at , but differentiable elsewhere on .

Therefore, the function is continuous on , but not differentiable on , particularly due to the point .