If I were to define a "function" mapping $\mathbb{R} \to \mathbb{R}$ such that $$f(x) = \begin{cases}\infty & x = 0\\ 0 & \text{else}\end{cases}$$ would this be consistent with definition of a function? (It is possible that this question comes down whether or not $\infty \in \mathbb{R}$ makes sense to write).
Is there anything about this mapping that indicates that it is NOT a function?
That is not a function $\mathbb{R} \to \mathbb{R}$ since $\infty \not\in \mathbb{R}$. It is a function $\mathbb{R} \to \bar{\mathbb{R}} = \mathbb{R} \cup \{\infty\}$ though.