Let $f$: $\mathbb{R}\longrightarrow\mathbb{R}$ be an arbitrary function.
Must there exist $E\subseteq\mathbb{R}$ of size continuum, such that $f$ restricted to $E$ is monotonic?
I guess this question can be generalized to functions $\kappa\longrightarrow\kappa$, with $\kappa$ infinite.
The answer is no. This follows from the following.
Claim (Sierpinski, Zygmund): There exists a function $f: \mathbb{R} \to \mathbb{R}$ such that for every $A \subseteq \mathbb{R}$, if $A$ has size continuum, then $f \upharpoonright A$ is not Borel.
Proof sketch: Construct $f$ such that for every real valued function $g$ with $dom(g)$ uncountable Borel, $\{x \in dom(g) : f(x) = g(x)\}$ has size less than continuum.