Solve the Following Questions


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 :

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

  2. $f(\mathrm{x})<0, \forall \mathrm{x} \in \mathrm{R}$

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

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

Correct Option: , 4


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

$x-y=h$ 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, $f(y)=1 \Rightarrow f(x)=1$

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