Does there exist any other function $xi$ that makes the function $f$ continuous on the set of real number $\mathbb R$? Let's define $\delta:\mathbb R\to \mathbb R$ as follows:
$\forall x\in\mathbb R,$ express $x$ as $x=7k+\delta$ with euclidean algorithm, where $\delta$ is the remainder and $7$ is the divisor. We associate $x$ with $\delta$. Thus finish our definition of $\delta$.
Let $\xi$ be a continuous function defined on $[0,7)$. Define $f(x)=4^{\frac{x-\delta}7}\xi(\delta)$. My question is: Apart from $\xi(x)=c*4^{\frac x7}$ where $c$ is a real number, does there exist any other function $\xi$ that makes the function $f$ increasing, or continuous, or differentiable on the set of real number $\mathbb R$?
 A: You are looking for functions $f$ with $f(x+7)=4f(x)$.  Any function $\xi$ from $[0,7)$ to $\mathbb{R}$ determines such a function $f$ from $\mathbb{R}$ to $\mathbb{R}$
The resulting $f$ is weakly increasing iff $\xi$ is and also $\xi(0)$ is greater than or equal to $1/4$ of the upper bound on $xi(x)$ as $x$ approaches 7. 
This function $f$ is continuous if and only if $\xi$ is continuous and the limit of $\xi(x)$ as $x$ approaches 7 is 4 times the value $f(0)$.  It is (once) differentiable if and only if it is continuous and furthermore $\xi$ is differentiable and the limit of the derivative $\xi'(x)$ as $x$ approaches 7 is four times the derivative $\xi'(0)$.
For $f$ to have higher derivatives impose the analogous condition on higher derivatives of $\xi$ as $x$ approaches 7.
All of these conditions allow $\xi$ infinitely much freedom between 0 and 7.
A challenging further step would be to describe explicitly all polynomial functions $\xi$, or all analytic functions $\xi$, giving $f$ these properties. Omran Kouba offers some analytic examples.
