Let $f : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous function such that $f(3x)-f(x)=x.$ If $f(8)=7$, then $f(14)$ is equal to

Question

Is there a general method to approach this other than $\lim_{ x\rightarrow \infty}$, as it is really difficult to think about that on spot.

Answer 1

$$f(x)-f\big(\frac{x}{3}\big)=\frac{x}{3}.$$ Similarly, $$f\big(\frac{x}{3^n}\big)-f\big(\frac{x}{3^{n+1}}\big)=\frac{x}{3^{n+1}}.$$

Let us add this up over all $n$.

This gives $f(x)=f(0)+\frac{x}{2}$ for any $x>0$,by continuity.

Since $f(8)=7$, $f(0)$ must be $3$. Therefore, $f(14)=7+3=10$.