A student writes:
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"I can't help noticing a similarity between the recursive definition of
a discrete function, and the derivative of a continuous function.
Both describe the change in the function (the recursive definition and
the derivative both tell you the change between one output and its
successor), and they both require an initial condition in order to find
the original equation (when integrating, you need the initial condition
to solve for the constant). In fact, the example you went over in class
followed the rules of a derivative: the function was linear, and its
recursive definition was a constant. Is the recursive definition of a
discrete function analogous to the derivative of a continuous function?
A better question might just be what is the difference between them?

Also, can y=sin(x) be represented by a discrete function?
If so, what is its recursive definition?"

Reply:
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Yes; they are not only similar, but they are intimately related.
The short reason is that, just as derivatives are differences between
values of a continuous function, the way we are computing (some)
recursively defined functions, they are what are called "finite"
differences. (That's why the 5 in 5n+1 is related to the 5 in h(n)+5.
Think about what the derivative of 5n+1 is.) For more on this topic,
take a look at Rosen, p. 460: read the paragraph following problem 61
(it has nothing to do with problem 61), and try (or at least read)
problems 62-68.

For more on difference equations (which are the discrete analog of
differential equations) and on their relationship to recursion and
recurrence relations, see:

http://en.wikipedia.org/wiki/Difference_equation#Relationship_to_difference_equations 

As for your question about sin(x), I think that one way of defining it
recursively would be to use something like the Taylor series expansion
for it (i.e., a way of computing sine in terms of standard arithemetical
operations and summations).

Good questions!