Number of functions f : 0 , 1 → 0 , 1 satisfying f x - f y = x - y for all x , y in 0 , 1 is

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Number of functions

f : [0, 1] \to [0, 1]

satisfying

|f (x) - f (y)| = |x - y|

for all

x, y

in

[0, 1]

is

MathematicsFunctionsKVPYKVPY 2018 (SB/SX)

Options:

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

Solution:

2561 Upvotes Verified Answer

The correct answer is: exactly

2

Given, $| f (x) - f (y) | = | x - y |$ for all in $[0, 1]$

$\Rightarrow \frac{| f (x) - f (y) |}{| x - y |} = 1$

$\Rightarrow \lim_{x \to y} \frac{| f (x) - f (y) |}{x - y} = 1 \Rightarrow |f^{'} (x)| = 1$

$\Rightarrow f^{'} (x) = \pm 1 \Rightarrow f (x) = \pm x$

$x \in [0, 1]$

$∴ f (x)$ has exactly $2$ function.

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