Answer to Question #250132 in Calculus for shantel

Question

Identify a mathematician that contributed to the development of theory of functions in Mathematics. State the contribution made and why was it so important to the field of Mathematics. 

Answer 1

The mathematical concept of a function emerged in the 17th century in connection with the development of the calculus; for example, the slope  of a graph at a point was regarded as a function of the x-coordinate of the point. Functions were not explicitly considered in antiquity, but some precursors of the concept can perhaps be seen in the work of medieval philosophers and mathematicians such as Oresme.

Mathematicians of the 18th century typically regarded a function as being defined by an analytic expression. In the 19th century, the demands of the rigorous development of analysis by Weierstrass and others, the reformulation of geometry in terms of analysis, and the invention of set theory by Cantor, eventually led to the much more general modern concept of a function as a single-valued mapping from one set to another.

Euler

In the first volume of his fundamental text Introductio in analysin infinitorum, published in 1748, Euler gave essentially the same definition of a function as his teacher Bernoulli, as an expression or formula involving variables and constants e.g., $x^2+3x+2$ . Euler's own definition reads:

A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities.^[15]

Euler also allowed multi-valued functions whose values are determined by an implicit equation.

In 1755, however, in his Institutiones calculi differentialis, Euler gave a more general concept of a function:

When certain quantities depend on others in such a way that they undergo a change when the latter change, then the first are called functions of the second. This name has an extremely broad character; it encompasses all the ways in which one quantity can be determined in terms of others.^[16]

Medvedev considers that "In essence this is the definition that became known as Dirichlet's definition." Edwards^[18] also credits Euler with a general concept of a function and says further that

The relations among these quantities are not thought of as being given by formulas, but on the other hand they are surely not thought of as being the sort of general set-theoretic, anything-goes subsets of product spaces that modern mathematicians mean when they use the word "function".