For the particular case of continuity, it seems to me that functions like $x \mapsto \begin{cases} x\sin\frac{1}{x} & x \neq 0 \\ 0 & x = 0\end{cases}$ and $x \mapsto \begin{cases} \sin\frac{1}{x} & x \neq 0 \\ 0 & x = 0\end{cases}$ are good motivators -- specifically, they both show that the naive definition doesn't always allow us to distinguish continuous from discontinuous functions. (I might hold an in-class vote about whether those two functions are continuous, for example -- I suspect that opinions would differ, and this would provide motivation for a definition that could be unambiguously tested.)