To analyze the problem provided in the image, let’s break down the information and examine each option:

Given Information:

• The function $f$ is continuous on $[0,1]$.
• $f(x) \in [0,1]$ for all $x \in [0,1]$.

Options:

1. (a) $f(x) = x$ for some $x \in [0,1]$:

   • Since $f(x) \in [0,1]$ and $x \in [0,1]$, it is possible for $f(x) = x$ to hold for some value of $x$ in $[0,1]$ without violating the conditions of continuity or the range restriction. This option could be true.

2. (b) $f(x) = x^2$ for all $x \in [0,1]$:

   • The function $f(x) = x^2$ satisfies continuity and maps $[0,1]$ to $[0,1]$. This function does not violate any of the given conditions, so this is also a valid candidate for $f(x)$.

3. (c) $f(x) = x - 1$ for all $x \in [0,1]$:

   • The function $f(x) = x - 1$ would map $[0,1]$ to $[-1,0]$, which is outside the specified range $[0,1]$ for $f(x)$. Therefore, this option cannot be true.

4. (d) None of the above:

   • This option would be true only if none of the options (a), (b), or (c) were valid.

Conclusion: Given the analysis:

• Option (c) is invalid.
• Both options (a) and (b) are possible under the given conditions.

Thus, the correct answer would be (d) None of the above, as (a) and (b) are valid under the constraints provided, but neither is definitive on its own.

Would you like more detailed explanations for any specific step or concept here?

Here are some related questions to explore:

1. Why is continuity important in determining the behavior of $f(x)$ on $[0,1]$?
2. How would the solution change if $f$ were only defined at discrete points within $[0,1]$?
3. What implications would arise if $f(x)$ were not constrained within $[0,1]$?
4. How would option (c) change if the interval were $[-1,0]$ instead?
5. Can we deduce anything further if $f$ is known to be differentiable?

Tip: Checking the range and continuity requirements for each option is crucial in problems involving constraints on functions within intervals.