Let `f:R rarr R` be such that `f(2x-1)=f(x)` for all `x in R`. If f is continuous at x = 1 and f(1) = 1. then

Options

A. `f(2)=1`
B. `f(2)=2`
C. f is continuous only at x = 1
D. f is continuous at all points

Answer

To solve the problem, we need to analyze the given functional equation and the continuity condition. Let's go through the solution step by step.

### Step 1: Understand the functional equation
We are given that \( f(2x - 1) = f(x) \) for all \( x \in \mathbb{R} \). This means that the function \( f \) has a specific symmetry or periodicity based on the transformation \( 2x - 1 \).

### Step 2: Substitute values into the functional equation
To explore the behavior of \( f \), we can substitute specific values for \( x \):
- Let \( x = 1 \):
\[ f(2 \cdot 1 - 1) = f(1) \implies f(1) = f(1) \]
This is trivially true.

- Let \( x = 0 \):
\[ f(2 \cdot 0 - 1) = f(0) \implies f(-1) = f(0) \]

- Let \( x = 2 \):
\[ f(2 \cdot 2 - 1) = f(2) \implies f(3) = f(2) \]

### Step 3: Explore further substitutions
We can continue substituting values to find more relationships:
- Let \( x = -1 \):
\[ f(2 \cdot (-1) - 1) = f(-1) \implies f(-3) = f(-1) \implies f(-3) = f(0) \]

- Let \( x = 3 \):
\[ f(2 \cdot 3 - 1) = f(3) \implies f(5) = f(3) \implies f(5) = f(2) \]

### Step 4: General pattern
From the substitutions, we observe that \( f(x) \) takes the same value for many different inputs. Specifically, we can see that:
- \( f(-1) = f(0) \)
- \( f(2) = f(3) = f(5) \)

This suggests that \( f \) may be a constant function.

### Step 5: Use continuity at \( x = 1 \)
We know that \( f \) is continuous at \( x = 1 \) and \( f(1) = 1 \). If \( f \) is constant, then it must be equal to 1 for all \( x \).

### Step 6: Conclude the nature of \( f \)
Since \( f \) is continuous at \( x = 1 \) and takes the value 1, and since we have shown that \( f \) takes the same value for many inputs, we conclude that:
\[ f(x) = 1 \quad \text{for all } x \in \mathbb{R} \]

### Step 7: Find \( f(2) \)
From our conclusion, we find:
\[ f(2) = 1 \]

### Step 8: Determine continuity
Since \( f(x) = 1 \) for all \( x \), it is continuous everywhere on \( \mathbb{R} \).

### Final Answer
Thus, we conclude that:
- \( f(2) = 1 \)
- \( f \) is continuous at all points in \( \mathbb{R} \).

---