Spiky Function continuity A function on an interval is continuous on that interval if and only if it is continuous on every point in that interval. Are "spiky" functions, defined on an interval, continuous? Such as $|x|$ on $\Bbb R$?

You can have corners in continuous functions, and the absolute value function is a good example. Clearly there is not problem except at $x=0$. For $x=0$ you can do an $\epsilon - \delta$ proof taking $\delta=\epsilon$ and show continuity.