The problem involves proving that a function f is constant given the condition |f(x) - f(y)| ≤ (x - y)² for all real numbers x and y. This falls within the subject area of mathematical analysis, particularly focusing on properties of functions and their continuity or differentiability.
The discussion is ongoing, with participants exploring different methods to approach the proof. There are multiple interpretations of how to utilize the condition provided, and hints have been shared to guide the reasoning process without reaching a consensus.
Some participants question the assumptions regarding the use of derivatives and the applicability of induction, indicating a need for clarity on the implications of the condition stated in the problem.
lurflurf said:Dick's approach is popular, but I never liked it. I do not think this problem merits the use of the derivative. I like to show inductively that
|f(x)-f(y)|<=(x-y)^2 implies
|f(x)-f(y)|<=(x-y)^2/2^n for any natural number n from which the result is obvious.
hint
|f(x)-f(y)|=|[f(x)-f((x+y)/2]+[f((x+y)/2)-f(y)]|<=|f(x)-f((x+y)/2|+|f((x+y)/2)-f(y)|