Real functions, Riemann integrals, and the differential operator are all introduced in calculus. When studying calculus we observe that both the integral and the differential operator act on real functions. We observe that for real functions, the differential operator will either create a new real function or nothing. If using the differential operator creates a new real function, we say that our original function is differentiable. For an example, look at Figure 1. The blue function is a result of applying the differential operator to the black function. Because applying the differential operator to our black function yields a new function, we say that the black function is differentiable.

For some real functions, the Riemann integral will produce a real number which may have a physical interpretation such as area or volume of a surface. Look again at Figure 1 for an example. The integral of our black function is 2 3 which can be interpreted to be the area between the black function and the x-axes on the interval [-1,1].

To give another example, let f be a real function defined on the interval [-1,1] and whose range is the set of all real numbers where f ( x ) = e - x 2 . Applying the integral to f on the interval [0,1] will yield a real number which we'll denote as 0 1 e - x 2 d x . To obtain a good approximation for 0 1 e - x 2 d x , we will do a Monte-Carlo simulation.