Mathematics - Continuity - Differentiability Question with Solution | TestHub

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

Let $x-y=h$

$\Rightarrow x=y+h$

$\underset{x\to 0}{\lim }\left|\frac{f\left(y+h\right)-f\left(y\right)}{h}\right|\le 0$

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

$\Rightarrow f\left(y\right)=k$   (constant)

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

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