Let be any function defined on R and let it satisfy the condition :
$|f (x) - f (y)| \leq | {(x - y)}^{2} |, \forall (x, y) \in R$ If $f (0) = 1,$ then

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

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

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

$f (x)$ can take any value

answer is A.

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Detailed Solution

$|\lim_{x \to y} \frac{f (x) - f (y)}{x - y}| \leq \underset{x \to y}{l i m} |x - y| \Rightarrow |f^{1} (y)| \leq 0, \forall y \Rightarrow f^{1} (x) = 0, \forall x$

$\Rightarrow f (x) = c f o r a l l x, \sin c e f (0) = 1 \Rightarrow c = 1$

f(x)= 1 $> 0$ for all x.

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