Which way of writing functions is the most correct?

Question

In functional programming it's not uncommon to bind a closure/lambda/anonymous function to a value name, i.e. $$f = x \mapsto x^2 + 3$$

so I've been wondering which is more right to do in mathematics?

$$FToC(f) = (f - 32) * \frac{5}{9}$$ or $$ FToC \colon f \mapsto (f - 32) * \frac{5}{9}$$

I mean in the end they both work the same way... it's $FToC(32) = 0$ in both cases

So is it just a matter of taste or is one actually more correct to use than the other?

Note that anonymous functions, or lambdas are sometimes also defined as so: $\lambda x.x+3$ which is effectively the same as $x \mapsto x+3$

For those not familiar, you can read more about lambdas here

Answer 1

Method A:

$f:\mathbb R\to\mathbb R$ is defined by $$f:x\mapsto (x-32)\cdots$$

Method B:

$f:\mathbb R\to\mathbb R$ is defined by $$f(x)=(x-32)\cdots$$

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In my experience it is common in pure mathematics to use either of the above notations, though the latter is more compact and seen as slightly less formal though no less "correct" - A is usually only used to really emphasize important definitions, whilst B will usually be used in examples etc.

In applied mathematics, B is overwhelmingly the most common notation, though I'm sure all actual mathematicians would understand the former too, though they may raise eyebrows.

Note that the statement of the domain and codomain should be present in any formal definition, though it is omitted where it is obvious or familiar.

Note also that "pure" and "applied" mathematics are both complete fictions.