newtheorem{What I use to summon the environment}{Name of the environment}

newtheorem{definition}[thm]{Definition}

begin{def}

Let $G$ be an open subset of $mathbb{C}$ and let $f: G to mathbb{C}$ be a
function. We say that $f$ is differentiable at $z in G$ if

$$lim_{h to 0} dfrac {f(z+h) - f(z)}{h}$$

exists. If it exists, we denote this limit as $f'(z)$.

end{def}

! Missing $ inserted.
<inserted text>
            $
l.49 ...n subset of $mathbb{C}$ and let $f: G to
                                               mathbb{C}$ be a function...