$f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^2$ , find $f(x)$ Find $f(x)$ if $f(x)-2f(\frac{x}{2})+f(\frac{x}{4})=x^2$, where $x, f(x)\in (-\infty , \infty)$ and $f(x)$ is continuous.
OK, since the hint seems not to be enough, here is the solution.

First,
Setting $g(x)=f(x)-f(\frac{x}{2})$ we have 
$$g(x)-g(\frac{x}{2})=x^2 \\
f(x)-f(\frac{x}{2})=g(x) \tag{*}$$

We solve the first equation.

By induction 
$$g(x)-g(\frac{x}{2^{n}})=x^2+\frac{x^2}{4}+\frac{x^2}{16}+...+\frac{x^2}{4^{n-1}}=x^2 \frac{1-\frac{1}{4^n}}{1-\frac{1}{4}}$$

Since $f$ is continuous at $x=0$, so is $g$. Taking the limit by $n$ you get
$$g(x)-g(0)=\frac{4}{3} x^2$$

Moreover, $f(x)-f(\frac{x}{2})=g(x)$ implies $g(0)=0$.

Now, we need to solve 
$$f(x)-f(\frac{x}{2})=\frac{4}{3}x^2$$
which is exactly the equation above, multiplies by $\frac{4}{3}$. Therefore, solving exactly as above we get
$$f(x)-f(0)=\frac{4}{3}\frac{4}{3}x^2=\frac{16}{9}x^2$$

This shows that 
$$f(x)=\frac{16}{9}x^2 +c$$
where $c=f(0)$ 
are all the solutions.

P.S. To make this more clear, the equation can be reduced via (*) to two equations of the type
$$h(x)-h(\frac{x}{2})=r(x)$$
with $h, r$ continuous. This equation can be solved as above: By induction
$$h(x)-h(\frac{x}{2^n})=\sum_{k=0}^{n-1}r(\frac{x}{2^n})$$
Using the continuity of $h$ at $x=0$ (we don't even need $h$ to be continuous at other points) we get
$$h(x)-h(0)=\sum_{k=0}^{\infty}r(\frac{x}{2^n})$$
(Note here that, if the series is divergent it implies that there is no solution which is defined at that $x$).

So, by calculating the series, you get the solution
$$h(x)=c+\sum_{k=0}^{\infty}r(\frac{x}{2^n})$$
where $c=h(0)$.
In this case, since the equation is linear and given the "clue" that the coefficient of $x^2$ is non-zero, then consider $f(x) = a + b x + c x^2$. This yields the possible polynomial solutions, but not necessarily all possible solutions.
\begin{align}
x^2 &amp;= (a + b x + c x^2) - 2 \, \left( a + \frac{b x}{2} + \frac{c x^2}{4} \right) +  \left( a + \frac{b x}{4} + \frac{c x^2}{16} \right) \\
&amp;= \frac{b x}{4} + \frac{ 9 \, c \, x^2}{16}.
\end{align}
This leads to $b=0$ and $c = 16/9$ and the general solution of the form
$$f(x) = a + \frac{16 \, x^2}{9}.$$