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