prove that every f: N→R is continuous?

Let f : N -> R be a function. Let n be a natural number and let e > 0. Set d = 1. Then for all m, |m - n| < d implies m = n, which implies |f(m) - f(n)| = 0 < e. Hence f is continuous at n. Since n was arbitrary, f is continuous.

the notion of continuity of a function on N

is not that obvious because N is a discrete set :

the first question to be answered is :

what is the vicinity of an integer ?

on other words, N must be defined with a topology,

so, a set of vicinity.

because the definition of continuity at a point xo is

for any x € vicinity of xo ==> f(x) € vicinity of f(xo)

maybe you should review the terms and rephrase.

hope it' ll help !!

michael