Mryntu

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.

As has been pointed out this definition is incorrect, as it is inconsistent with the usual definitions of continuity and differentiability. Your example $|x|$ suffices to show this.

If you are encountering this in multivariable calculus then your professor might be trying to state the theorem mentioned by avs in the comments: that a function is differentiable at a point if all its first order partial derivatives exist in a neighbourhood of that point, and are continuous at that point. However the converse is not generally true: consider for example the function
$$f(x,y)=begincases(x^2+y^2)sin(frac1sqrtx^2+y^2) &(x,y)neq(0,0)\0&(x,y)=(0,0)endcases$$
at the origin. Thus this assumption might be completely false. It might be best to give a word for word reproduction of the statement and the paragraph before and after.

Sounds like the wrong definition of what a "continuous function" is. Any function $f:mathbbRtomathbb R$ like in your original post is continuous at every point $left(a,fleft(aright)right)$ wherever $$limlimits_xto a^-fleft(xright)=limlimits_xto a^+fleft(xright)$$ (denoting the left and right-hand limits accordingly)