Is $f(z)= |z|$ continuous on the complex plane? So I understand that the absolute value of $z=a+b\mathbf i=\sqrt{a^2+b^2}$, I just don't know if its enough to just say that this is continuous so $f$ is continuous or if I have to go through an epsilon-delta proof. A brief explanation of the structure of the proof would be greatly appreciated. 
Thank you for any help!

It is standard to use $\epsilon$-argument to deduce that $h(x,y)=x^{2}+y^{2}$ is continuous on the plane, now use the fact that composition with continuous map is still continuous to conclude that $f$ is continuous: $\xi(u)=\sqrt{u}$ and $f=\xi\circ h$.
The continuity of a complex-valued function is the same issue with the continuity of the corresponding two variables function: $f(z)=|z|=f(x,y)=\sqrt{x^{2}+y^{2}}$.
