Let f be any function defined on R and let it satisfy the condition : [|f(x) - f(y)|, ,|(x…

Question

Let f be any function defined on R and let it satisfy the condition : $$|f(x) - f(y)|\, \le \,|{(x - y)^2}|,\forall (x,y) \in R$$

If f(0) = 1, then :

Accepted Answer

$$|f(x) - f(y)|\, \le \,|{(x - y)^2}|$$

$$ \Rightarrow \left| {{{f(x) - f(y)} \over {x - y}}} \right|\, \le \,|x - y|$$

$$ \Rightarrow \left| {\mathop {\lim }\limits_{x \to y} {{f(x) - f(y)} \over {x - y}}} \right|\, \le \,|\mathop {\lim }\limits_{x \to y} (x - y)|$$

$$ \Rightarrow |f'(x)|\, \le 0$$

$$ \Rightarrow f'(x) = 0$$

$$ \Rightarrow f(x)$$ is constant function.

$$ \because $$ $f(0)$ = 1 then f(x) = 1

Solutions · 1