# Let f be any function defined on R

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:

1. (1) $f(x)<0, \forall x \in R$

2. (2) $f(x)$ can take any value in $R$

3. (3) $f(x)=0, \forall x \in R$

4. (4) $f(x)>0, \forall x \in R$

Correct Option: , 4

Solution:

$|f(x)-f(y)| \leq\left|(x-y)^{2}\right|, \forall(x, y) \in R$

$\left|\frac{f(x)-f(y)}{x-y}\right| \leq|x-y|$

$\lim _{x \rightarrow y}\left|\frac{f(x)-f(y)}{x-y}\right| \leq 0$

$\left|f^{\prime}(y)\right| \leq 0 \Rightarrow f^{\prime}(y)=0$

$f(y)=C$

$\Rightarrow \quad c=1$

$\Rightarrow f(x)=1$