MathDB

Prove that any continuos function

f: \mathbb{R}\rightarrow \mathbb{R}

with f(x)\equal{}\left\{ \begin{aligned} a_1x\plus{}b_1\ ,\ \text{for } x\le 1 \\ a_2x\plus{}b_2\ ,\ \text{for } x>1 \end{aligned} \right. where

a_1,a_2,b_1,b_2\in \mathbb{R}

, can be written as: f(x)\equal{}m_1x\plus{}n_1\plus{}\epsilon|m_2x\plus{}n_2|\ ,\ \text{for } x\in \mathbb{R} where

m_1,m_2,n_1,n_2\in \mathbb{R}

and \epsilon\in \{\minus{}1,\plus{}1\}.

Problem Statement