real analysis - Continuity of a ruler function.

A question from Introduction to Analysis by Arthur Mattuck:

Define the "ruler function" $f(x)$ as follows:

$$f(x)=\begin{cases} {1/2^n}, & \text{if $x={b/2^n}$ for some odd integer $b$;} \\ 0, & \text{otherwise}. \end{cases}$$

(a) Prove that $f(x)$ is discontinuous at the points $b/2^n$, ($b$ odd).

(b) Prove $f(x)$ is continuous at all other points.

I prove (a) by constructing a sequence {$x_n$} of irrational numbers whose limit is $b/2^n$. Then the limit of {$f(x_n)$} is $0$, since $f(x_n)=0$ for all $n$. But $f(b/2^n)\ne 0$, discontinuity occurs. I don't know how to prove (b).