solve this equation :
f(t) =f(2t) +f(2t-1)
and for t<1 :f(t)=0

We are given the equation:

\[
f(t) = f(2t) + f(2t - 1)
\]

and the boundary condition \( f(t) = 0 \) for \( t < 1 \).

### Step 1: Analyze for \( t \geq 1 \)

For \( t \geq 1 \), we need to explore the recursive nature of the function. We will substitute values for \( t \) and see how the recursion plays out.

#### Case 1: \( t = 1 \)
\[
f(1) = f(2 \times 1) + f(2 \times 1 - 1)
\]
This simplifies to:
\[
f(1) = f(2) + f(1)
\]
Subtracting \( f(1) \) from both sides:
\[
0 = f(2)
\]

#### Case 2: \( t = 2 \)
\[
f(2) = f(2 \times 2) + f(2 \times 2 - 1)
\]
This simplifies to:
\[
f(2) = f(4) + f(3)
\]
But from the previous case, we know \( f(2) = 0 \), so:
\[
0 = f(4) + f(3)
\]
Thus, \( f(4) = -f(3) \).

### Step 2: Extend the recursion

From these initial cases, we can observe that the values of the function depend recursively on higher arguments. However, since we know \( f(t) = 0 \) for \( t < 1 \), and since \( f(2) = 0 \), we may conjecture that the function remains zero for all \( t \geq 1 \).

### Step 3: Hypothesis

Given that every time we substitute into the recursion, we eventually hit \( f(t) = 0 \), it's plausible that \( f(t) = 0 \) for all \( t \geq 1 \). Therefore, the solution to the functional equation is:

\[
f(t) = 0 \quad \text{for all} \ t
\]

### Conclusion
The function \( f(t) \) is identically zero for all \( t \).

---

Would you like to explore details about recursion, functional equations, or the implications of this result?

Here are 5 related questions:
1. How do we solve other functional equations involving recursion?
2. Can non-trivial solutions exist in similar recursive equations?
3. What are the properties of functions defined by recurrence relations?
4. How do boundary conditions influence the behavior of recursive functions?
5. How can such functional equations be applied in real-world problems?

**Tip:** In recursive equations, boundary conditions often play a crucial role in determining whether non-zero solutions exist.

Explore the solution to the functional equation f(t) = f(2t) + f(2t - 1) with detailed step-by-step analysis. Understand the recursive nature and boundary conditions involved in this mathematical problem. Suitable for advanced high school students and anyone interested in functional equations and recursion.

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