Up until now, we have dealt with sequences -- functions from the natural numbers to real numbers. Now we turn our attention to functions from real numbers to real numbers, so-called "real functions".
A real function is a function from a set, A to the real numbers, \mathbb{R}. Usually A will be an "interval" -- all numbers between two numbers, all numbers less than a number, all numbers greater than a number or all numbers.
We will usually keep A implicit, and assume it contains anything we are interested in.
If f and g are functions, f+g is the function defined by (f+g)(x)=f(x)+g(x). Similarly, we define fg, -f and f-g. If f is a function, everywhere that f(x)\ne 0, we can define 1/f.
We note that with these definitions, functions on a given set A obey the so-called ring axioms.
The simplest example of a function is f(x)=a, the constant function. The next-simplest example of a function is f(x)=x, the identity function.
With these two functions,
Examples