Let $I$ be the subset $[-1,1]$ of $\Bbb R$. Consider a function $f:I\to I$ with properties:
(a) $f(x) = 0$ at all points $x_n = 2^{1-n}$ and $y_n = -2^{1-n}$ in $I$ ($n$ varies throughout all natural numbers).
(b) $f(x)$ is continuous at all these points.
(c) $f(x)$ is a bounded function.
(d) For all integers $n$, $f(x)$ with domain restricted to $[-2^{-n},2^{-n}]$ is equal to $\dfrac 1 {a^n}f(2^nx)$ for some fixed constant $a > 1$.
(e) $f(0) = 0.$
Is $f(x)$ necessarily continuous at the point $x = 0$?