Mathematics
Exploring Cauchy’s Functional Equation
How do we solve such a problem? First, as always, it is a good idea to plug in some numbers
Now, we start to see a pattern and we try to make a guess:
We will use mathematical induction to prove that it holds for all natural n
And thus we have shown that our original assumption was true.
Now let’s continue plugging stuff in:
Finally, we replace x = m/n and f(1) = c (we can do that because f(1) is clearly a constant) and get the desired result
Are there any generalizations?
The functional equation we have just solved is called Cauchy’s functional equation and it is one of the famous results in Olympiad Algebra.
We have solved it over rational numbers and you might wonder, what about the set of all real numbers ℝ? To answer this question we will state a theorem (which we will not prove):
Conclusion
Cauchy’s functional equation might seem simple, but it has more useful applications than you might think. We will solve Olympiad-level problems that use Cauchy’s functional equation in future articles, so stay tuned.
