The given problem appears to ask for a proof that under certain conditions, there exists a point $\xi \in [a, b]$ such that $f(\xi) = 0$. Let’s solve it step by step.

Problem Statement

Given:

1. $f(x) \in C[a, b]$, meaning $f(x)$ is continuous on $[a, b]$,
2. For any $x \in [a, b]$, there exists $y \in [a, b]$ such that $|f(y)| \leq \frac{1}{2}|f(x)|$,

Prove: There exists $\xi \in [a, b]$ such that $f(\xi) = 0$.

Proof

1. Use the property of continuity

Since $f(x) \in C[a, b]$, it is bounded and attains its maximum and minimum on $[a, b]$. Let: $M = \max_{x \in [a, b]} |f(x)|$ Clearly, $M \geq 0$.

2. Assume $M > 0$ and derive a contradiction

Suppose $M > 0$. Then there exists $x_0 \in [a, b]$ such that $|f(x_0)| = M$. Using the given property, for this $x_0$, there exists $y \in [a, b]$ such that: $|f(y)| \leq \frac{1}{2}|f(x_0)| = \frac{1}{2}M.$

3. Apply the iterative argument

By repeating the same argument for $y$, there exists some $z \in [a, b]$ such that: $|f(z)| \leq \frac{1}{2}|f(y)| \leq \frac{1}{4}M.$ Continuing this process, we construct a sequence $(x_n)$ such that: $|f(x_n)| \leq \frac{1}{2^n}M.$

4. Conclude using continuity

As $n \to \infty$, $\frac{1}{2^n}M \to 0$. By the continuity of $f(x)$, this implies there exists some $\xi \in [a, b]$ such that: $f(\xi) = 0.$

5. Final statement

Thus, we have shown that if $M > 0$, a contradiction arises, proving that $M = 0$ or $f(\xi) = 0$ for some $\xi \in [a, b]$.

Conclusion

There exists $\xi \in [a, b]$ such that $f(\xi) = 0$.

Let me know if you'd like further clarification or details. Here are five related questions for further exploration:

1. How does the intermediate value theorem relate to this proof?
2. Can this result be generalized for discontinuous functions?
3. What happens if $f(x)$ is not bounded on $[a, b]$?
4. How does the hypothesis $|f(y)| \leq \frac{1}{2}|f(x)|$ contribute to the proof?
5. Can we construct an explicit example of such a function $f(x)$?

Tip: When working with continuity-based proofs, always leverage key properties like boundedness and the intermediate value theorem for critical arguments.