Question:
Let $f$ be any function defined on $\mathrm{R}$ and let it satisfy the condition : $|f(x)-f(y)| \leq\left|(x-y)^{2}\right|, \forall(x, y) \in R$
If $f(0)=1$, then:
Correct Option: , 4
Solution:
$|f(x)-f(y)| \leq\left|(x-y)^{2}\right|, \forall(x, y) \in R$
$\left|\frac{f(x)-f(y)}{x-y}\right| \leq|x-y|$
$\lim _{x \rightarrow y}\left|\frac{f(x)-f(y)}{x-y}\right| \leq 0$
$\left|f^{\prime}(y)\right| \leq 0 \Rightarrow f^{\prime}(y)=0$
$f(y)=C$
$\Rightarrow \quad c=1$
$\Rightarrow f(x)=1$