** Discrete Function vs Continuous Function **

Functions are one of the most important classes of mathematical objects, which are extensively used in almost all sub fields of mathematics. As their names suggest both discrete functions and continuous functions are two special types of functions.

A function is a relation between two sets defined in such a way that for each element in the first set, the value that corresponds to it in the second set is unique. Let *f* be a function defined from the set *A* into set *B*. Then for each x*ϵ A, *the symbol *f*(x) denotes the unique value in the set *B* that corresponds to x. It is called the image of x under *f*. Therefore, a relation *f* from A into B is a function, if and only if for, each *xϵ A *and* y ϵ A; *if *x = y* then *f*(x)* = f*(y). The set A is called the domain of the function *f, *and it is the set in which the function is defined.

For example, consider the relation *f* from R into R defined by *f*(x) = x + 2 for each *xϵ A*. This is a function whose domain is R, as for each real number x and y, x = y implies *f*(x) = x + 2 = y + 2 = *f*(y). But the relation *g* from N into N defined by *g*(x) = a, where ‘a’ is a prime factors of x is not a function as *g*(6)=3, as well as *g*(6)=2.

**What is a discrete function?**

A discrete function is a function whose domain is at most countable. Simply, this means that it is possible to make a list that includes all the elements of the domain.

Any finite set is at most countable. The set of natural numbers and the set of rational numbers are examples for at most countable infinite sets. The set of real numbers and the set of irrational numbers are not at most countable. Both the sets are uncountable. It means that it is impossible to make a list that includes all the elements of those sets.

One of the most common discrete functions is the factorial function. *f* :N U{0}→N recursively defined by *f*(n) = n*f*(n-1) for each n ≥ 1 and *f*(0)=1 is called the factorial function. Observe that its domain N U{0} is at most countable.

**What is a continuous function? **

Let *f* be a function such that for each k in the domain of *f*, *f*(x)→*f*(k) as x → k. Then *f*is a continuous function. This means that it is possible to make *f*(x) arbitrarily close to *f*(k) by making x sufficiently close to k for each k in the domain of *f. *

Consider the function *f*(x) = x + 2 on R. It can be seen that as x → k, x + 2 → k + 2 that is *f*(x)→*f*(k). Therefore, *f *is a continuous function. Now, consider *g *on positive real numbers *g*(x) = 1 if x > 0 and *g*(x) = 0 if x = 0. Then, this function is not a continuous function as the limit of *g*(x) does not exist (and hence it is not equal to *g*(0) ) as x → 0.** **

• A discrete function is a function whose domain is at most countable but it need not be the case in continuous functions. • All continuous functions ƒ have the property that ƒ(x)→ƒ(k) as x → k for each x and for each k in the domain of ƒ, but it is not the case in some discrete functions. |