Are the functions with the map $\sum a_n b^{-n}\mapsto\sum a_n c^{-n}$ continuous? 
Let $(a_n)$ a sequence such that $0\le a_n<\min\{b,c\}$ and $b,c> 1$. Then define the function
$$f:A\to[0,1],\quad \sum_{n=1}^\infty a_n b^{-n}\mapsto\sum_{n=1}^\infty a_n c^{-n}$$
where $A$ is the subset of $[0,1]$ where the function is defined.

My question is: any map of this kind is continuous? I have a partial answer for the case of the Cantor function but I dont know if the continuity is true for the general case stated above.
If it is true the continuity of these maps, can you show any proof or hint for some proof? Thank you.

For concreteness, consider the case where $b=4, c=5$.  Then take any point $\mathbf{a}$ with a terminating quaternary expansion; for instance, $\mathbf {a}=\frac{9}{16}$, which corresponds to the sequence $\langle a_n\rangle = \langle 2, 1, 0, 0, 0, 0, \ldots\rangle$.  Then the sequence of sequences $$\begin{align}
\alpha_n&=\langle 2, 0, 3, 0, 0, 0, \ldots\rangle\\
\beta_n&=\langle 2, 0, 3, 3, 0, 0, \ldots\rangle\\
\gamma_n&=\langle 2, 0, 3, 3, 3, 0, \ldots\rangle\\
\end{align}
$$
etc. clearly converges to $\mathbf{a}$ from beneath, but the sequence $f(\mathbf{\alpha}), f(\mathbf{\beta}), f(\mathbf{\gamma})$, etc. converges to $\frac25 + \frac0{25}+\frac{3}{125}+\frac{3}{625}+\ldots$ $=\frac{2}{5}+\frac{3}{100}$ $=\frac{43}{100}$ $\neq f(\mathbf{a}) = \frac{2}{5}+\frac{1}{25} = \frac{44}{100}$, so $f()$ isn't continuous from below.  In fact, this is a general phenomenon; $f()$ won't be continuous at any value with a terminating base-$b$ expansion, if $b\lt c$.
