Question

The number of functions $f:[0,1]\to [0,1]$ satisfying $|f\left(x\right)–f\left(y\right)|=|x–y|$ for all $x,y$ in $[0,1]$ is

• A. exactly 1
• B. exactly 2
• C. more than 2, but finite
• D. infinite

Answer 1

The correct option is B exactly 2

Given :
$|f\left(x\right)–f\left(y\right)|=|x–y|$
Taking limit,
$\underset{y\to x}{lim}\frac{|f\left(x\right)-f\left(y\right)|}{|x-y|}=1$
$\Rightarrow \underset{y\to x}{lim}\left|\frac{f\left(x\right)-f\left(y\right)}{x-y}\right|=1$
$\Rightarrow \underset{y\to x}{lim}\left|\frac{f\left(y\right)-f\left(x\right)}{y-x}\right|=1$
$\Rightarrow |f^{\prime }\left(x\right)|=1\Rightarrow f^{\prime }\left(x\right)=\pm 1$
$\Rightarrow f\left(x\right)=\pm x+C$

As the function $f:[0,1]\to [0,1]$



Hence there are only two possible solution of $f\left(x\right)$
$f\left(x\right)=x$ and $f\left(x\right)=1-x$