Why is 'abuse of notation' tolerated? I've personally tripped up on a few concepts that came down to an abuse of notation, and I've read of plenty more on stack exchange.  It seems to all be forgiven with a wave of the hand.  Why do we tolerate it at all?
I understand if later on in one's studies if things are assumed to be in place, but there are plenty of textbooks out there assuming certain things are known before teaching them.  This is a very soft question, but I think it ought to be asked.

Done properly abusing notation makes things clearer. Suppose that $f \colon X \rightarrow Y$ and $A \subseteq X$. We can write


*

*$f(A)$

*$\{ f(x) \colon x \in A \} $

*Define $g \colon X \cup \mathcal{P}(X) \rightarrow Y \cup \mathcal{P}(Y)$ by $g(x) = f(x)$ if $x \in X$ and $g(A) = \{ f(x) \colon x \in A \} $ if $A \subseteq X$. Here we can use both $g(x)$ and $g(A)$. There are difficulties with this method if $X \cap \mathcal{P}(X) \neq \varnothing$.

