I'll first state the question:
Let $f:[0,1] \to [0,1]$ be a function defined as follows: $f(1)=1$, and if $a=0.a_1a_2a_3\ldots$ is the decimal representation of a (which does not end with a chain of 9's), then $f(a)=0.0a_10a_20a_3\ldots$ . Discuss the continuity of $f$ at $0.392$ .
Progress and observations so far:
Clearly $f:\sum_{i=0}^{\infty}( \frac{a_i}{10^i}) \mapsto \sum_{i=0}^{\infty}( \frac{a_i}{10^{2i}})$ . Take $a=0.a_1a_2a_3a_4\ldots=0.392$ . If I choose $x=0.x_1x_2x_3\ldots$ from a sufficiently small interval on the right side of $a$, say $[a,0.3929]$, then it can be shown that $|f(x)-f(a)|=f(x)-f(a)=\sum_{i} \frac{x_i-a_i}{10^{2i}} \leq \sum_{i} \frac{x_i-a_i}{10^{i}}=|x-a|$ since for all $i\in \mathbb{N}$ $a_i \leq x_i$ and thus $\frac{x_i-a_i}{10^{2i}} \leq \frac{x_i-a_i}{10^{i}}$. This proves that $\lim_{x\to a+}f(x)=f(a)$.
The troubles I'm facing:
I'm not very sure about the above method and I can't think on a better one. Also, by the above method I can't prove $\lim_{x\to a-}f(x)=f(a)$. As the main question uses the word "discuss", I'm not sure whether it really asks to prove or disprove the continuity of $f$ at $a$.
My questions are:
1) Is the function $f$ continuous at $a=0.392$?
2) What can be said about the continuity of $f$ at any arbitrary point from $[o,1]$ rather than the particular point $a=0.392$?
3) Is there any way to view this map graphically?