Finding a continuous non-monotonic functions First, I would like to find a function $f:\left[ 0,1 \right] \rightarrow \mathbb{R}$ such that $f$ is continuous but not monotonic in any interval. Secondly, I want to find a, continuous yet not monotonic in any interval, function $g:\mathbb{R} \rightarrow \mathbb{R} $.
Since a function is not continuous in any interval $\leftrightarrow$ the function is not continuous in any interval with rational edges, I thought about finding a function similar to dirichlet function, but couldn't find one that would suit me. Any suggestions? Or maybe I'm looking for the wrong "type" of functions? 

If a function is monotonic on an interval it is differentiable almost everywhere on that interval, so it suffices to find a continuous nowhere differentiable function.  One such example is the Takagi function.  This gives you the function $f$ you are looking for.  For the function $g$ just paste together a bunch of copies of $f$.  
