In the following, [x] denotes the greatest integer less than or equal to `x`.
`{:(,"Column I",,"Column II"),(A,x|x|,p,"continuous in (-1, 1)"),(B,sqrt(|x|),q,"differentiable in (-1, 1)"),(C,x + [x],r,"strictly increasing (-1, 1)"),(D,|x-1| + |x + 1|", in"(-1,1),s,"not differentiable atleast at one point in (-1, 1)"):}`

Answer

To solve the problem, we need to analyze each function given in Column I and determine its properties as described in Column II. We will go through each function step by step.

### Step 1: Analyze Function A: \( f(x) = |x| \)

1. **Continuity in (-1, 1)**:
- The function \( f(x) = |x| \) is continuous everywhere, including the interval (-1, 1).
- **Conclusion**: **True** (Option P)

2. **Differentiability in (-1, 1)**:
- The function \( f(x) = |x| \) is not differentiable at \( x = 0 \) because the left-hand derivative and right-hand derivative do not match.
- **Conclusion**: **False** (Option Q)

3. **Strictly Increasing (-1, 1)**:
- The function is not strictly increasing because it has a minimum at \( x = 0 \).
- **Conclusion**: **False** (Option R)

4. **Not Differentiable at least at one point in (-1, 1)**:
- As established, it is not differentiable at \( x = 0 \).
- **Conclusion**: **True** (Option S)

### Summary for Function A:
- Matches: P (True), S (True)

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### Step 2: Analyze Function B: \( f(x) = \sqrt{|x|} \)

1. **Continuity in (-1, 1)**:
- The function \( f(x) = \sqrt{|x|} \) is continuous everywhere, including (-1, 1).
- **Conclusion**: **True** (Option P)

2. **Differentiability in (-1, 1)**:
- The function is not differentiable at \( x = 0 \) because the left-hand derivative and right-hand derivative do not match.
- **Conclusion**: **False** (Option Q)

3. **Strictly Increasing (-1, 1)**:
- The function is not strictly increasing because it has a minimum at \( x = 0 \).
- **Conclusion**: **False** (Option R)

4. **Not Differentiable at least at one point in (-1, 1)**:
- As established, it is not differentiable at \( x = 0 \).
- **Conclusion**: **True** (Option S)

### Summary for Function B:
- Matches: P (True), S (True)

---

### Step 3: Analyze Function C: \( f(x) = x + [x] \)

1. **Continuity in (-1, 1)**:
- The function \( f(x) = x + [x] \) is not continuous at integer points (specifically at \( x = 0 \)).
- **Conclusion**: **False** (Option P)

2. **Differentiability in (-1, 1)**:
- The function is not differentiable at \( x = 0 \) because it has a jump discontinuity.
- **Conclusion**: **False** (Option Q)

3. **Strictly Increasing (-1, 1)**:
- The function is strictly increasing in the interval (-1, 1).
- **Conclusion**: **True** (Option R)

4. **Not Differentiable at least at one point in (-1, 1)**:
- As established, it is not differentiable at \( x = 0 \).
- **Conclusion**: **True** (Option S)

### Summary for Function C:
- Matches: R (True), S (True)

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### Step 4: Analyze Function D: \( f(x) = |x - 1| + |x + 1| \)

1. **Continuity in (-1, 1)**:
- The function \( f(x) = |x - 1| + |x + 1| \) is continuous everywhere, including (-1, 1).
- **Conclusion**: **True** (Option P)

2. **Differentiability in (-1, 1)**:
- The function is not differentiable at \( x = -1 \) and \( x = 1 \) because the slopes change at these points.
- **Conclusion**: **False** (Option Q)

3. **Strictly Increasing (-1, 1)**:
- The function is not strictly increasing because it is constant between the points where it changes slope.
- **Conclusion**: **False** (Option R)

4. **Not Differentiable at least at one point in (-1, 1)**:
- As established, it is not differentiable at \( x = -1 \) and \( x = 1 \).
- **Conclusion**: **True** (Option S)

### Summary for Function D:
- Matches: P (True), S (True)

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### Final Matching:
- **A**: P, S
- **B**: P, S
- **C**: R, S
- **D**: P, S