The number of non differentiability of point of function `f (x) = min ([x] , {x}, |x - (3)/(2)|)` for `x in (0,2),` where [.] and {.} denote greatest integer function and fractional part function respectively.

Answer

To determine the number of points of non-differentiability of the function \( f(x) = \min([x], \{x\}, |x - \frac{3}{2}|) \) for \( x \in (0, 2) \), we will analyze each component of the function separately and then find the points where the function is not differentiable.

### Step 1: Analyze the components of the function

1. **Greatest Integer Function \([x]\)**:
- This function is constant on intervals of the form \([n, n+1)\) where \(n\) is an integer.
- In the interval \((0, 2)\), \([x]\) takes the values:
- \([x] = 0\) for \(0 \leq x < 1\)
- \([x] = 1\) for \(1 \leq x < 2\)
- It is non-differentiable at \(x = 1\) (jump discontinuity).

2. **Fractional Part Function \(\{x\}\)**:
- This function is defined as \(\{x\} = x - [x]\) and is continuous and differentiable everywhere except at integer points.
- In the interval \((0, 2)\), \(\{x\}\) takes the values:
- \(\{x\} = x\) for \(0 \leq x < 1\)
- \(\{x\} = x - 1\) for \(1 \leq x < 2\)
- It is non-differentiable at \(x = 1\).

3. **Absolute Value Function \(|x - \frac{3}{2}|\)**:
- This function is linear except at the point \(x = \frac{3}{2}\) where it has a sharp corner.
- In the interval \((0, 2)\), it is defined as:
- \(|x - \frac{3}{2}| = \frac{3}{2} - x\) for \(0 < x < \frac{3}{2}\)
- \(|x - \frac{3}{2}| = x - \frac{3}{2}\) for \(\frac{3}{2} < x < 2\)
- It is non-differentiable at \(x = \frac{3}{2}\).

### Step 2: Determine the points of non-differentiability

Now we need to find the points where the minimum function \(f(x)\) is non-differentiable.

- From the analysis:
- \([x]\) is non-differentiable at \(x = 1\).
- \(\{x\}\) is non-differentiable at \(x = 1\).
- \(|x - \frac{3}{2}|\) is non-differentiable at \(x = \frac{3}{2}\).

### Step 3: Identify the overall points of non-differentiability

- At \(x = 1\), both \([x]\) and \(\{x\}\) contribute to non-differentiability.
- At \(x = \frac{3}{2}\), \(|x - \frac{3}{2}|\) contributes to non-differentiability.

Thus, the points of non-differentiability of \(f(x)\) in the interval \((0, 2)\) are:
1. \(x = 1\)
2. \(x = \frac{3}{2}\)

### Conclusion

The total number of points of non-differentiability of the function \(f(x)\) in the interval \((0, 2)\) is **2**.

---