This is a homework question from Complex Variables by Joseph L. Taylor. The questions reads as follows:
Prove that if $f$ is a continuous function defined on an open subset $U$ of $\mathbb{C}$, then sets of the form {$z\in{U}$:$|f(z)|<r$} and {$z\in{U}$: Re$(f(z))<r$} are open.
Here is what I have:
Let $f$ be a continuous function and $U$ be an open, non-empty subset of $\mathbb{C}$. Let $a\in{U}$. Since $f$ is continuous, $\forall \epsilon>0$, $\exists \delta>0$ such that, if $|z-a|<\delta$ then $|f(z)-f(a)|<\epsilon$. Let $D$ be a neighborhood around $a$. If $z\in{D_{\delta}(a)}\implies f(z)\in{D_{\epsilon}(f(a))}$. Then $f(D_{\delta}(a))\subseteq D_{\epsilon}(f(a))$. So, $D_{\delta}(a)\subseteq f^{-1}(D_{\epsilon}(f(a)))$.
Here is where I am having a hard time, connecting where I am at to {$z\in{U}$:$|f(z)|<r$} and {$z\in{U}$: Re$(f(z))<r$}. I spoke with my professor and he says that I need to show how $f^{-1}(D_{\epsilon}(f(a)))\subseteq f^{-1}(D_r(0))$, i.e. I have one property left to prove. But I am seriously drawing a blank.
I drew this picture to help me visualize what is happening, but I feel like I am missing pieces in it, or it is not drawn properly. 
Any help will be much appreciated. And to be honest, if you feel there are improvements that can be made to how I write proofs I am always open to suggestions for improvement.
The functions $g:x\rightarrow |x|$ and $h:x\rightarrow Re(x)$ are continue, thus $(g\circ f)^{-1}(0,r)$ and $(h\circ f)^{-1}(0,r)$ are open.