If `f (x)= [{:(x,,, "if x is rational"), (1-x,,," if x is irrational "):},` then number of points for `x in R,` where `y =f (f (x))` discontinous is:

Options

A. (a) 0
B. (b) 1
C. (c) 2
D. (d) Infinitely many

Answer

To solve the problem, we need to analyze the function \( f(x) \) and then find \( f(f(x)) \) to determine where it is discontinuous.

### Step 1: Define the function \( f(x) \)
The function \( f(x) \) is defined as:
- \( f(x) = x \) if \( x \) is rational (denoted as \( x \in \mathbb{Q} \))
- \( f(x) = 1 - x \) if \( x \) is irrational (denoted as \( x \in \mathbb{R} \setminus \mathbb{Q} \))

### Step 2: Find \( f(f(x)) \)
We will evaluate \( f(f(x)) \) for both cases of \( x \).

#### Case 1: \( x \) is rational
If \( x \) is rational, then:
\[ f(x) = x \]
Now, we need to find \( f(f(x)) \):
\[ f(f(x)) = f(x) = f(x) = x \quad (\text{since } x \text{ is rational}) \]

#### Case 2: \( x \) is irrational
If \( x \) is irrational, then:
\[ f(x) = 1 - x \]
Now, we need to find \( f(f(x)) \):
\[ f(f(x)) = f(1 - x) \]
Since \( 1 - x \) is also irrational (as the sum of a rational and an irrational number is irrational), we have:
\[ f(1 - x) = 1 - (1 - x) = x \]

### Step 3: Combine results
From both cases, we find:
\[ f(f(x)) = x \quad \text{for all } x \in \mathbb{R} \]

### Step 4: Determine continuity
The function \( f(f(x)) = x \) is a linear function, which is a polynomial function. Polynomial functions are continuous everywhere in their domain.

### Conclusion
Since \( f(f(x)) \) is continuous for all \( x \in \mathbb{R} \), there are no points of discontinuity.

Thus, the number of points where \( y = f(f(x)) \) is discontinuous is:
\[ \boxed{0} \]