If ` f (x) = {{:((1- sqrt2sin x )/(pi - 4 x )",",, x ne (pi)/(4)),(a" "",",,x = (pi)/(4)):}`
is continuous at ` x = (pi)/(4)`, then a =

Options

A. 4
B. 2
C. 1
D. `1//4`

Explanation

To determine the value of \( a \) that makes the function \( f(x) \) continuous at \( x = \frac{\pi}{4} \), we need to ensure that the limit of \( f(x) \) as \( x \) approaches \( \frac{\pi}{4} \) is equal to \( f\left(\frac{\pi}{4}\right) \).

### Step 1: Define the function
The function is defined as:
\[ f(x) = \begin{cases} \frac{1 - \sqrt{2} \sin x}{\pi - 4x} & \text{if } x \neq \frac{\pi}{4} \\ a & \text{if } x = \frac{\pi}{4} \end{cases} \]

### Step 2: Find the limit as \( x \) approaches \( \frac{\pi}{4} \)
We need to calculate:
\[ \lim_{x \to \frac{\pi}{4}} f(x) = \lim_{x \to \frac{\pi}{4}} \frac{1 - \sqrt{2} \sin x}{\pi - 4x} \]

### Step 3: Substitute \( x = \frac{\pi}{4} \) into the limit
First, we calculate \( \sin\left(\frac{\pi}{4}\right) \):
\[ \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \]
Now substitute \( x = \frac{\pi}{4} \):
\[ \lim_{x \to \frac{\pi}{4}} f(x) = \frac{1 - \sqrt{2} \cdot \frac{\sqrt{2}}{2}}{\pi - 4 \cdot \frac{\pi}{4}} = \frac{1 - 1}{\pi - \pi} = \frac{0}{0} \]
This is an indeterminate form, so we can apply L'Hôpital's rule.

### Step 4: Apply L'Hôpital's Rule
Differentiate the numerator and denominator:
- The derivative of the numerator \( 1 - \sqrt{2} \sin x \) is \( -\sqrt{2} \cos x \).
- The derivative of the denominator \( \pi - 4x \) is \( -4 \).

Now we can rewrite the limit:
\[ \lim_{x \to \frac{\pi}{4}} \frac{-\sqrt{2} \cos x}{-4} = \frac{\sqrt{2}}{4} \cos\left(\frac{\pi}{4}\right) \]
Substituting \( \cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \):
\[ \lim_{x \to \frac{\pi}{4}} f(x) = \frac{\sqrt{2}}{4} \cdot \frac{\sqrt{2}}{2} = \frac{2}{8} = \frac{1}{4} \]

### Step 5: Set the limit equal to \( a \)
For \( f(x) \) to be continuous at \( x = \frac{\pi}{4} \), we need:
\[ a = \lim_{x \to \frac{\pi}{4}} f(x) = \frac{1}{4} \]

### Conclusion
Thus, the value of \( a \) that makes the function continuous at \( x = \frac{\pi}{4} \) is:
\[ \boxed{\frac{1}{4}} \]