Using "implies" to refer to material conditional

Is it acceptable to translate the binary connective "$\let\ f\rightarrow$" into English with "implies"? I'm unsure because "implies" for me immediately brings to mind logical implication, but I've seen some places use it for the material conditional (including wikipedia, in the opening sentence of this article).

For example, does mathematical convention, in principle, permit the following formulation of the standard definition for functional continuity?

$f$ is continuous at $c$ if for any $\epsilon > 0 :$ there exists $\delta > 0:$ for any $x$ in Domain[$f$]$:$ $|x-c|<\delta$ implies $|f(x)-f(c)|<\epsilon$.

• $\huge\rightarrow$ (the material conditional) is a logical connective that operates on two sentences $P$ (the antecedent) and $Q$ (the consequent) to form a new sentence. The resultant truth function $$P\rightarrow Q$$ is neutral to the truth values of $P$ and $Q$, and is false precisely when $P$ is true but $Q$ false.

• $\huge\Rightarrow$ (consequence/implication) also connects an antecedent $P$ and a consequent $Q,$ but rather than operating on them per se, it draws an inference: the resultant meta-sentence $$P\Rightarrow Q$$ asserts that the sentence $\,P\rightarrow Q\,$ is true. Some ways to read $\,\large P\Rightarrow Q\normalsize:$

  • If $P$ (is true), then $Q$ (is true).
  • $P$ (being true) is a sufficient condition for $Q$ (to be true).
  • $P$ (being true) implies (that) $Q$ (is true).
  • $P$ (is true) only if $Q$ (is true).
  • $Q$ (being true) is a necessary condition for $P$ (to be true).

  If $\,P\Rightarrow Q\,$ is false, then it must be that $P$ is true yet $Q$ false.

$\large P\rightarrow Q\,$ is not an inferential sentence: it does not generally mean $$\text{‘}P \text{ being true implies that } Q \text{ is true’.}$$ As such, I refrain from calling $\,\rightarrow\,$ “material implication”. When reading $\large\,P\rightarrow Q\,$ as $\text{‘if } P \text{ then } Q\text{’},$ bear in mind that it is truth-functional—not an assertion of truth.

To be clear: $\,P\rightarrow Q\,$ is synonymous with $\,P\Rightarrow Q\,$ only when the former is actually true.

In the given formulation $$|x-c|<\delta\;\; \textbf{implies} \;\;|f(x)-f(c)|<\epsilon,$$ “implies” is presumably a translation of the implication symbol $\large\Rightarrow\normalsize,$ not the material conditional $\large\rightarrow\normalsize;$ in that case the original sentence does read coherently.

P.S. I distinguish between implication $\,\large\Rightarrow\,$ and logical implication $\,\large\vDash\,,$ which is often used to mean first-order implication/consequence, i.e., that $\,P\rightarrow Q\,$ is true regardless of interpretation.

P.P.S. Symbolic logic is an area rife with conflicting notation, terminology and even notions; my understanding is eclectically evolving.

Yes.

By the way, logical implication is material conditional. In logic only the forms of the arguments matter in order to deduce from something.

When you see $$ |x-c|<\delta\text{ implies }|f(x)-f(x)|>\epsilon $$ written in a proof, it is certainly an English version of the formal statement $$ |x-c|<\delta\to|f(x)-f(x)|>\epsilon. $$

Sometimes authors say at the beginning of their book that the proofs will be given in an informal manner. Informal means that English language will be used for better readability. In principle, those informal proofs could be made formal in, say, first-order logic.

Note. The logical connective $\to$ really contains what we mean by "implies". Indeed, $p\to q$ does what it is supposed to do: it permits us to infer $q$ from $p$ but nothing from $\neg p$.