We want to find all continuous functions $f:R→R$ that satisfy the equation $f(x^2+1/4)=f(x)$ for all real x.
Of course -If I am right- constant functions satisfy the equation mentioned, and as well many other do not such as polynomials, rational functions and etc.; But to find all of them!
I need your hints or solutions with my regards.
Answer
From $f(-x)=f((-x)^2+\frac14)=f(x)$ we see that $f$ is even and it suffices to consider $x\ge 0$.
For $x_1\in[0,\frac12)$, the sequence defined by $x_{n+1}=x_n^2+\frac14$ is remains within that interval and is strictly increasing, hence converges to the fixpoint $\frac12$. By the property and continuity we conclude $f(x_1)=f(x_2)=f(x_3)=\ldots=f(\frac12)$.
For $x_1>\frac12$, the sequence defined by $x_{n+1}=\sqrt{x_n-\frac14}$ remains in $(\frac12,\infty)$, is stictly decreasing, hence converges to the fixpoint $\frac12$. Again we conclude $f(x_1)=f(x_2)=f(x_3)=\ldots=f(\frac12)$.
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