Let f be any function defined on R and let it | vedclass

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Question

$\left|\frac{f(x)-f(y)}{(x-y)}\right| \leq|(x-y)|$

$x-y=h$ let $\Rightarrow x=y+h$

$\lim _{x \rightarrow 0}\left|\frac{f(y+h)-f(y)}{h}\right| \l

Vedclass · Accepted Answer

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

Vedclass · Answer

$\left|\frac{f(x)-f(y)}{(x-y)}\right| \leq|(x-y)|$

$x-y=h$ let $\Rightarrow x=y+h$

$\lim _{x \rightarrow 0}\left|\frac{f(y+h)-f(y)}{h}\right| \leq 0$

$\Rightarrow\left|f^{\prime}( y )\right| \leq 0 \Rightarrow f^{\prime}( y )=0$

$\Rightarrow f( y )= k ($ constant $)$

and $f(0)=1$ given

So, $f(y)=1 \Rightarrow f(x)=1$

Let $f$ be any function defined on $R$ 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

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