Let f(x) be a continuous function, `AA x in R, f(0) = 1 and f(x) ne x` for any `x in R`, then show `f(f(x)) gt x, AA x in R^(+)`

Answer

To solve the problem, we need to show that \( f(f(x)) > x \) for all \( x \in \mathbb{R}^+ \) under the conditions given: \( f(0) = 1 \) and \( f(x) \neq x \) for any \( x \in \mathbb{R} \).

### Step-by-Step Solution:

1. **Understanding the Function**:
Given that \( f(0) = 1 \), we can start by analyzing the behavior of \( f(x) \) at \( x = 0 \). Since \( f(0) = 1 \), we know that the function passes through the point (0, 1).

**Hint**: Consider the implications of the function value at \( x = 0 \) and how it relates to the line \( y = x \).

2. **Behavior of \( f(x) \)**:
We know that \( f(x) \neq x \) for any \( x \in \mathbb{R} \). This means the graph of \( f(x) \) does not intersect the line \( y = x \).

**Hint**: Think about what it means for a function to not equal \( x \) and how this affects the graph's position relative to the line \( y = x \).

3. **Analyzing the Graph**:
Since \( f(0) = 1 \) and \( f(x) \neq x \), if we assume \( f(x) < x \) for some \( x \), then \( f \) would have to cross the line \( y = x \) at some point, which contradicts our condition. Therefore, \( f(x) > x \) for \( x > 0 \).

**Hint**: Use the properties of continuous functions and the Intermediate Value Theorem to reason about the behavior of \( f(x) \).

4. **Substituting \( f(x) \)**:
Now, since we have established that \( f(x) > x \) for all \( x > 0 \), we can substitute \( f(x) \) into itself:
\[ f(f(x)) > f(x) \]
because \( f(x) > x \).

**Hint**: Remember that if \( a > b \), then applying a function that is increasing (like \( f \) in this case) will maintain the inequality.

5. **Combining Inequalities**:
From our previous steps, we have:
\[ f(f(x)) > f(x) > x \]
Thus, we can conclude that:
\[ f(f(x)) > x \]

**Hint**: When you have a chain of inequalities, you can combine them to show the final result.

6. **Conclusion**:
Therefore, we have shown that \( f(f(x)) > x \) for all \( x \in \mathbb{R}^+ \).

### Final Statement:
Thus, we conclude that \( f(f(x)) > x \) for all \( x \in \mathbb{R}^+ \) under the given conditions.

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