"Analytic functions are those that can be represented by a power series, convergent within a certain region bounded by the so-called circle of convergence. Outside of this region the analytic function is not regarded as given a priori ; its continuation into wider regions remains a matter of special investigation and may give very different results, according to the particular case considered." (Felix Klein, "Sophus Lie", [lecture] 1893)
"Logic sometimes makes monsters. For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose. More of continuity, or less of continuity, more derivatives, and so forth. Indeed, from the point of view of logic, these strange functions are the most general; on the other hand those which one meets without searching for them, and which follow simple laws appear as a particular case which does not amount to more than a small corner. In former times when one invented a new function it was for a practical purpose; today one invents them purposely to show up defects in the reasoning of our fathers and one will deduce from them only that. If logic were the sole guide of the teacher, it would be necessary to begin with the most general functions, that is to say with the most bizarre. It is the beginner that would have to be set grappling with this teratologic museum." (Henri Poincaré, 1899)
"In order to regain in a rigorously defined function those properties that are analogous to those ascribed to an empirical curve with respect to slope and curvature (first and higher difference quotients), we need not only to require that the function is continuous and has a finite number of maxima and minima in a finite interval, but also assume explicitly that it has the first and a series of higher derivatives (as many as one will want to use)." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)
"The course of the values of a continuous function is determined at all points of an interval, if only it is determined for all rational points of this interval."
"The most general definition of a function that we have reached in modern mathematics starts by fixing the values that the independent variable x can take on. We define that x must successively pass through the points of a certain 'point set'. The language used is therefore geometric […]." (Felix Klein, "Elementary Mathematics from a Higher Standpoint" Vol III: "Precision Mathematics and Approximation Mathematics", 1928)
"The modern definition of a function in the context of real numbers is that it is a relationship, usually a formula, by which a correspondence is established between two sets A (the domain) and B (the range) of real numbers in such a manner that to every number in set A there corresponds only one number in set B." (Alan Jeffrey, "Mathematics for Engineers and Scientists", 1969)
"If you start with a number and form its square root, you get another number. The term for such an 'object' is function. You can think of a function as a mathematical rule that starts with a mathematical object-usually a number-and associates to it another object in a specific manner. Functions are often defined using algebraic formulas, which are just shorthand ways to explain what the rule is, but they can be defined by any convenient method. Another term with the same meaning as 'function' is transformation: the rule transforms the first object into the second. […] Operations and functions are very similar concepts. Indeed, on a suitable level of generality there is not much to distinguish them. Both of them are processes rather than things."
"If each change of a certain quantity results in a corresponding change of another quantity, we can say that there exists a functional relationship between those two quantities. Viewed in this manner, the idea of functions expands endlessly. The concept of functions is truly comprehensive, but while it is all encompassing, it is not fathomless; at least, not with respect to our current subject of manifolds. You might feel that linear functions or quadratic functions are far too specific and that you are sinking into the depths of the ocean called functions. However, you will be rescued from the ocean depths by understanding of the functions that are needed to describe manifolds. These functions are continuous, analytic, and differentiable functions." (Kenji Ueno & Toshikazu Sunada, "A Mathematical Gift, III: The Interplay Between Topology, Functions, Geometry, and Algebra", Mathematical World Vol. 23, 1996)
"I turn away with fright and horror from the lamentable evil of functions which do not have derivatives." (Charles Hermite, [letter to Thomas J Stieltjes])
"When graphing a function, the width of the line should be inversely proportional to the precision of the data." (Marvin J Albinak)
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