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The problem provides a function \( f: (0, +\infty) \to \mathbb{R} \) that satisfies the equation:

\[
f^3(x) + f(x) = \ln(x), \quad \text{for all } x \in (0, +\infty).
\]

It asks us to:

**(a)** Find the value of \( f(1) \).  
**(b)** Prove that the function \( f \) is continuous at \( x_0 = 1 \).

---

### Part (a): Finding \( f(1) \)

We substitute \( x = 1 \) into the given equation:

\[
f^3(1) + f(1) = \ln(1).
\]

Since \( \ln(1) = 0 \), the equation simplifies to:

\[
f^3(1) + f(1) = 0.
\]

Factoring this expression gives:

\[
f(1) \left( f^2(1) + 1 \right) = 0.
\]

The term \( f^2(1) + 1 \) is always positive, so the only solution is \( f(1) = 0 \).

---

### Part (b): Continuity at \( x_0 = 1 \)

To prove that \( f \) is continuous at \( x_0 = 1 \), we need to show that:

\[
\lim_{x \to 1} f(x) = f(1).
\]

Since we know that \( f(1) = 0 \), we need to investigate the limit of the function as \( x \to 1 \).

Starting with the original equation \( f^3(x) + f(x) = \ln(x) \), we aim to understand the behavior of \( f(x) \) near \( x = 1 \).

1. Near \( x = 1 \), \( \ln(x) \) is close to \( \ln(1) = 0 \).
2. Thus, for \( x \) near 1, the function equation becomes approximately \( f^3(x) + f(x) \approx 0 \), implying that \( f(x) \approx 0 \), confirming that \( \lim_{x \to 1} f(x) = 0 \).

Since \( \lim_{x \to 1} f(x) = f(1) = 0 \), the function \( f \) is continuous at \( x_0 = 1 \).

---

Would you like more details on any step or additional clarifications?

### Relative questions:
1. What are the properties of the function \( f \) for \( x > 1 \) or \( x < 1 \)?
2. How do you solve similar functional equations involving powers and logarithms?
3. Can you use limits to explore differentiability, beyond continuity, at \( x_0 = 1 \)?
4. What other types of problems involve proving continuity?
5. How can the behavior of \( \ln(x) \) as \( x \to 0^+ \) influence the form of \( f(x) \)?

### Tip:
When analyzing continuity, always check if the function's limit at the point of interest matches its value at that point.

Δίνεται η συνάρτηση f: (0, +∞) → ℝ τέτοια ώστε f³(x) + f(x) = ln(x), για κάθε x ∈ (0, +∞). Να βρείτε την τιμή f(1) και να αποδείξετε ότι η f είναι συνεχής στο x₀ = 1.

Learn how to solve the functional equation f³(x) + f(x) = ln(x) for the value of f(1) and prove that the function f is continuous at x₀ = 1. This problem covers functional equations, logarithms, and continuity concepts. Suitable for undergraduate mathematics students.

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