Let f be any function defined on R and let it satisfy the...

Let $x - y = h$

$⇒ x = y + h$

$\lim_{x \to 0}\left|\dfrac{f\left(y + h\right) - f\left(y\right)}{h}\right| \le 0$

$⇒ \left|{f}^{'}\left(y\right)\right| \le 0 ⇒ {f}^{'}\left(y\right) = 0$

$⇒ f\left(y\right) = k$ (constant)

And $f\left(0\right) = 1$ given

So, $f\left(y\right) = 1 ⇒ f\left(x\right) = 1$