Confusion about non-derivable continuous functions I am reading a definition which claims that a function is continuous in point $p$ iff all its first derivations exist and are continuous in the point $p$. And what confuses me are functions such as $f(x)=|x|$ which should be continuous by intuition, but is clearly not derivable in $x=0$. I am almost certain I am getting something wrong here, but I can not even pin-point what.

That "definition" is wrong. You are right, the function $|x|$ is continuous but is not differentiable at $x=0$. Continuity doesn't imply differentiability. However, **differentiability does imply continuity**.

The definition you stated looks to me as an attempt to define a smooth function, although it is not correct.