One of the surest sources of confusion in mathematical writing is to use one expression to mean multiple things. Authors often make this mistake with functions. To illustrate, consider a function $f$ from $\mathbb{R}$ to $\mathbb{R}$. The expressions “$f$” and “$f(x)$” have different meanings. The statement “$f(x)$ is monotonically decreasing” is meaningless because “$f(x)$” is a constant real number. Instead, write “$f$ is monotonically decreasing” or “$x \mapsto f(x)$ is monotonically decreasing”. The latter form is useful if you want to talk about the composition of functions without introducing a new symbol: “$x \mapsto f(1 + x^3)$ is monotonically decreasing”.

When is it OK to reuse a symbol?

A complete ban on reusing symbols in a mathematical text would cause you to quickly run out of symbols, especially in long texts. Similar to variables in a programming language, symbols have a particular scope.

The tightest scope is what I will call a bound symbol. A bound symbol is defined and used only in a single expression. The symbol $x$ is a bound variable in the following expressions: “$\forall x\ P(x)$”, “$\exists x\ P(x)$”, and “$x \mapsto f(x)$.” It is OK to immediately use the same bound symbol in multiple expressions, such as, “Consider the functions $x \mapsto f(x)$ and $x \mapsto g(x)$.” To aid the reader, however, each time a bound symbol is reused in a section of text, it should have the same type of meaning.

Symbols introduced in a proof should be restricted to that proof (many readers skip proofs on their first read of a text), so we can treat a proof as a symbol scope. If we introduce “$\varepsilon > 0$ and $\delta = 2\varepsilon$” in the proof of Theorem 1, then it is OK to define “$\delta \in (0, 1)$ and $\varepsilon = \sqrt{\delta}$” in the proof of Theorem 2.

Symbols introduced in definitions and theorems are not scoped in the same way.