Differentiability

The function involves absolute values, necessitating careful consideration of different intervals for continuity and differentiability. First, evaluate it for various cases:

For :

and . Thus, .

For :

and . Thus, .

For :

and . Thus, .

The piecewise function becomes:

Continuity:

Check continuity at critical points , .

At :

, , and , so is continuous.

At :

, , and , so is continuous.

Differentiability:

Check differentiability at critical points , .

For , derivative .

For , derivative .

For , derivative .

At :

Left-hand derivative .

Right-hand derivative .

Since left-hand and right-hand derivatives are not equal, is not differentiable at .

At :

Left-hand derivative .

Right-hand derivative .

Since left-hand and right-hand derivatives are not equal, is not differentiable at .

Conclusion: The function is continuous everywhere but differentiable at all points except . Thus, the correct answer is:

Continuous everywhere and differentiable at all points except at .

Step 1: Simplify .
The given equation is:
.
Using logarithmic properties:
.
Step 2: Simplify .
The given equation is:
.
Using :
.
Step 3: Differentiate with respect to .
.
Step 4: Differentiate with respect to .
.
Step 5: Compute .
.
Simplify:
.
Using the property , rewrite:
.
Final Answer: