`f(x) = {{:(1 + cos^(-1){cot x} " ," x < pi/2),(pi[x] + 1 " ," x ge pi/2):}`
Where [ ] denotes greatest integer and { } denotes fractional part function, Then value of jump of discontinuity is:

Options

A. `pi/2`
B. `pi/2 - 1`
C. `0`
D. `1 - pi/2`

Explanation

To find the value of the jump of discontinuity for the given piecewise function \( f(x) \), we will evaluate the function at the point \( x = \frac{\pi}{2} \) from both the left and the right.

### Step 1: Define the function
The function is defined as:
\[ f(x) = \begin{cases} 1 + \{ \cos^{-1}(\cot x) \} & \text{if } x < \frac{\pi}{2} \\ \pi [x] + 1 & \text{if } x \geq \frac{\pi}{2} \end{cases} \]
where \( [x] \) denotes the greatest integer function and \( \{x\} \) denotes the fractional part function.

### Step 2: Evaluate the right limit as \( x \) approaches \( \frac{\pi}{2} \)
For \( x \geq \frac{\pi}{2} \):
\[ f\left(\frac{\pi}{2} + h\right) = \pi \left[\frac{\pi}{2} + h\right] + 1 \]
As \( h \to 0 \), \( \left[\frac{\pi}{2} + h\right] = 1 \) (since \( \frac{\pi}{2} \approx 1.57 \)). Thus:
\[ \lim_{h \to 0^+} f\left(\frac{\pi}{2} + h\right) = \pi \cdot 1 + 1 = \pi + 1 \]

### Step 3: Evaluate the left limit as \( x \) approaches \( \frac{\pi}{2} \)
For \( x < \frac{\pi}{2} \):
\[ f\left(\frac{\pi}{2} - h\right) = 1 + \{\cos^{-1}(\cot(\frac{\pi}{2} - h))\} \]
Using the identity \( \cot\left(\frac{\pi}{2} - h\right) = \tan(h) \), we have:
\[ \lim_{h \to 0^+} f\left(\frac{\pi}{2} - h\right) = 1 + \{\cos^{-1}(\tan(h))\} \]
As \( h \to 0 \), \( \tan(h) \to 0 \) and thus \( \cos^{-1}(0) = \frac{\pi}{2} \). Therefore:
\[ \lim_{h \to 0^+} f\left(\frac{\pi}{2} - h\right) = 1 + \left\{\frac{\pi}{2}\right\} = 1 + \left(\frac{\pi}{2} - 1\right) = \frac{\pi}{2} \]

### Step 4: Calculate the jump of discontinuity
The jump of discontinuity is given by the difference between the right limit and the left limit:
\[ \text{Jump} = \lim_{h \to 0^+} f\left(\frac{\pi}{2} + h\right) - \lim_{h \to 0^+} f\left(\frac{\pi}{2} - h\right) \]
Substituting the limits we calculated:
\[ \text{Jump} = (\pi + 1) - \left(\frac{\pi}{2}\right) \]
Simplifying this:
\[ \text{Jump} = \pi + 1 - \frac{\pi}{2} = \frac{2\pi}{2} + 1 - \frac{\pi}{2} = \frac{\pi}{2} + 1 \]

### Final Answer
The value of the jump of discontinuity is:
\[ \frac{\pi}{2} + 1 \]