Enderton Elements of Set Theory, p. 43 (1977, Academic Press), writes:
There was a reluctance to separate the concept of a function itself from the idea of a written formula defining the function.
What is the basis for the above historical claim? And at around what point did the concept of a function itself from the idea of a formula become firmly separated?
It seems interesting that what is today regarded as an elementary mistake had a strong historical basis.
Fuller quote from Enderton:
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The "father" of modern mathematical logic, Gottlob Frege assumed in his system the concept of function as primitive, and the concept of set (interpreted as the extension of a function whose values are only truth-values) is in some way derived from that of function.
We can say with a little pun that all Category Theory is the "revenge" of function over "subset of the cartesian product".
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I'm not sure it would be accurate to say there was "reluctance", in the sense of deliberate opposition to the idea, but it's true that many authors in the 18th century thought of functions as algebraic expressions, which was natural considering those were the only functions they'd ever seen.
All of my information comes from Kline's Mathemtical Thought.
As far as I know it is wrong to say that the notion of a function should be attributed to Euler. Euler introduced the notation $f(x)$ for a function in 1734, However, even he shouldn't be given full credit, as in 1718 Bernouilli already used $\phi x$, and as far as I knew Euler knew Bernouilli well, so it seems the only thing we really owe him is the use of $f$ and of parentheses. The word function is supposedly due to Leibniz as early as 1673. The wealth of notations would seem to indicate that the function concept was already important in 18th century work.
Kline states that in the 18th century it was "on the whole" believed that functions must be given by some sort of symbolic expression. He mentions that Euler and Lagrange called functions "discontinuous" if they were given by different expressions in different domains, presumably what we would call piece-wise defined functions. There are a few references (Gauss, Lacroix) from the very late 18th century defining functions more generally and more explicitly, in the case of Lacroix: "Every quantity whose value depends on one or several others (...) whether one knows or one does not know by what operations it is necessary to go from the latter to the first quantity" and gives the example of a root of a quintic as a function of its coefficients.
Fourier went out of his way in his Analytical Theory of Heat (1822) to explain that a function does not have to be expressed by any type of formula.
Cauchy was one of the earliest researchers to attempt to rigorously and systematically found analysis as far as I know. His Cours d'Analyse of 1821 is based on the fairly ambiguous concept of a "variable". He states that a variable is called a function of another variable if it is possible to determine its value from the other variable. But despite this slightly vague definition, he then proceeds to use the concept of a function in a fairly modern way, writing things like "Let $f(x)$ be a function of the variable $x$..." and define a notion of continuity for functions, for example. Cauchy's notion of "variable" was attacked by Weierstrass some years later (mid 19th century), saying that phrases such as "a variable approaches a limit" suggest some vague notion of time and motion. Weierstass is credited with the fully modern definition of a limit using $\epsilon-\delta$.
Already in the 1820s Dirichlet gave the definition that a function is an association of a unique value to each possible input value. The Dirichlet function ($1$ for irrationals, $0$ for rationals) is something he came up with in 1829, I would imagine as an example of how general his notion of function was, but Kline doesn't mention specifically.
In any case, Kline's description gives the impression that although the modern definition was definitely starting to emerge, there was no real consensus. Some used Dirichlet's definition, others assumed the function had to be given by "some law". To me this indicates that textbook authors broadly didn't consider it necessary to give an utterly precise definition, and followed their intuition.
If you're interested, this information is specifically in volume 3 of Mathematical Thought, p. 949.
The notion of a function is generally attributed by historians to Euler, in the 18th century. This notion was of course tied in with the idea of an algebraic expression or more general formula or what might today be called a definable" function, in a sense to be specified. A more general notion of a function as an arbitrary relation, i.e. a subset of the Cartesian product of the domain and the range, is due to Dirichlet, in the 19th century. This point of view eventually won over but there has been a persistent minority view opposing it, represented for example by Kronecker, Brouwer, Poincare, and others. This minority view is associated with what is known as intuitionism or constructivism. Some of these issues are discussed in this article in Intellectica.