Let $f_1:[0,1]\to\mathbb{R}$, $f_1(0)=0$ be continuously differentiable function and $\lambda>1$. Consider the sequence of function defined by $f_k(x):=\lambda f_{k-1}(x/\lambda)$, $k≥2,$ , $x\in [0,1]$. Find the pointwise limit of the sequence of function.
My Approach:
By the recurring relation, we get, $f_n(x)=\lambda^{n-1} f_1(x/\lambda)$. If we take the limit and using the continuity of $f_1$, we come across a situation of $\infty •0$. Please help me further.
$f_n(x)=\lambda^{n-1} f_1(x/\lambda)$ is not correct. We have $f_n(x)=\lambda^{n-1} f_1(x/\lambda^{n-1}).$
Can you proceed ?