I’d like to use this as part of a proof, but I couldn’t realize how to show this (and if it) is true. The function is: f(x)=x+(1+ex)−1

**Answer**

f is differentiable at R, then by MVT,

f(a)−f(b)=(a−b)f′(c)

with a<c<b.

but f′(c)=1−ec(1+ec)2=1+ec+e2c(1+ec)2

⟹0<f′(c)<1

thus |f(a)−f(b)|<|a−b|

your statement is not true for f.

**Attribution***Source : Link , Question Author : user8011332 , Answer Author : hamam_Abdallah*