Suppose we have $(\mathbb{R}, \tau_{st})$, standard topological space on $\mathbb{R}$. Then modulus function between topological spaces $f: \mathbb{R} \rightarrow \mathbb{R}, x \mapsto |x|$ is not continuous.
Proof: $$\begin{align} f^{-1}((-1,1)) & = \{x \in \mathbb{R}: f(x) \in (-1,1)\} \\ &= \{x \in \mathbb{R}: |x| \in (-1,1)\} \\ &= [0,1) \end{align}$$
But $[0,1)$ is not open in $\mathbb{R}$, hence $f$ is not continuous. $\square$
Clearly this proof is incorrect since $f$ is continuous in the usual metric sense, but I can't find the mistake.