Discuss whether the given function is smooth, piecewise smooth, continuous, piecewise continuous, or none of these on the interval $\left [ -\pi ,\pi \right ]$
$f(x)=\left \{ \begin{matrix} 1 & \mbox{if }x\mbox{ is irrational} \\ 0 & \mbox{if }x\mbox{ is rational }\end{matrix}\right. $
I have trouble advancing with this problem, though it seemed quite straight forward at glance.
It is discontinuous at any point, since for every point $x$ and for any neighbirhood you choose of $x$, for $\epsilon=0.5$, there is some element in that neighborhood, $y$, s.t $|x-y|>\epsilon$ (try understanding who $y$ is for yourself), and so $f$ does not satisfy the condition for continuity in any point.
Since it is discontinuous at any point, it naturally cannot be smooth, or any of the other options.
This function, by the way, is related to the Dirichlet function - a very important counter-example in analysis.