Let f:[0,1] → R be such that f(x y)=f(x). f(y), for all x, y ∈[0,1], and f(0) ≠ 0. If y=y(x) satisfies the differential equation, (d y)/(d x)=f(x)…

Let $\displaystyle f:[0,1] \rightarrow \mathbf{R}$ be such that $\displaystyle f(x y)=f(x) . f(y)$, for all $\displaystyle x, y \in[0,1]$, and $\displaystyle f(0) \neq 0$. If $\displaystyle y=y(x)$ satisfies the differential equation, $$\frac{d y}{d x}=f(x) \text { with } y(0)=1 \text {, then } y\left(\frac{1}{4}\right)+y\left(\frac{3}{4}\right) $$ is equal to :