To determine the correct statements about the function \(f(x) = \left(\frac{|x|}{2} - x\right)\left(x - \frac{|x|}{2}\right)\), we need to analyze the function's behavior, its derivative, and continuity. Let's proceed step-by-step:

1. Understanding the Function:

The given function is \(f(x) = \left(\frac{|x|}{2} - x\right)\left(x - \frac{|x|}{2}\right)\). Notice the use of absolute value, which suggests the function might behave differently for different ranges of \(x\):

• For \(x \geq 0\), \(|x| = x\).
• For \(x < 0\), \(|x| = -x\).

2. Case Analysis:

• Case 1: \(x \geq 0\)
  Substituting \(|x| = x\), the function becomes: 

• 
  This is a downward-opening parabola with its vertex (maximum point) at \(x = 0\).
• Case 2: \(x < 0\)
  Substituting \(|x| = -x\), the function becomes: 

• 
  This is also a downward-opening parabola with its vertex (maximum point) at \(x = 0\).

3. Derivative and Differentiability:

Calculate the derivative \(f'(x)\) for each of the cases:

• For \(x \geq 0\):
   

• For \(x < 0\):
   

We need to verify the derivative at \(x = 0\) and ensure continuity:

• From the right: 

• From the left: 

Both side derivatives approach the same limit as \(x \to 0\), indicating \(f'(x)\) is continuous at \(x = 0\). However, \(f(x)\) is non-differentiable at \(x = 0\) due to the change in behavior.

Conclusion:

• The function \(f(x)\) has a local maximum at \(x = 0\).
• The derivative \(f'(x)\) is continuous over \(\mathbb{R}\) but not differentiable at \(x = 0\).

Therefore, the correct statements are:

• \(f\) has a local maximum.
• \(f'\) is continuous over \(\mathbb{R}\).
• \(f'\) is not differentiable over \(\mathbb{R}\).