On Nov 5, 2012, at 2:59 PM, "Dave L. Renfro" <firstname.lastname@example.org> wrote:
> You > can also have '5' as a function, namely the constant function > f(x) = 5. In the well known (in mathematics) book "Rings of > Continuous Functions" by Gillman and Jerison, constant functions > are represented by bold face numerals. Thus, in that book, > a bold face '5' means the constant function f(x) = 5, whereas > a regular face '5' means the number. I believe Karl Menger > also used numerals to represent constant functions in his 1940s > and 1950s attempts to reform the teaching of elementary calculus. > > Functions represented by numerals can get more exotic than > this. For instance, when solving differential equations using > algebraic operator methods (google "differential equations" > along with "D operator"), numerals now represent multiplication > by constant operators on sets of functions. For example, 'D + 5' > represents the operation "d/dx + 5", which when you input > the function f(x), outputs the function f'(x) + 5*f(x). > In this setting, '5' would then represent a function whose > domain is a certain set of real-valued functions of one real > variable (all functions differentiable on a specified interval, > for example) and whose range is a similar (but not necessarily > the same) set of functions, which is defined by "5 evaluated > at f(x)" is equal to 5*f(x).