Question 73
Prove that any continuous function $f : \mathbb{R} \to \mathbb{R}$ of the form
$$
f(x) = \begin{cases} a_1x + b_1, & \text{for } x \le 1 \\ a_2x + b_2, & \text{for } x \gt 1 \end{cases}
$$
with $a_1, a_2, b_1, b_2 \in \mathbb{R}$, can be represented as
$$
f(x) = m_1x + n_1 + \varepsilon|m_2x + n_2|, \text{ for } x \in \mathbb{R},
$$
where $m_1, m_2, n_1, n_2 \in \mathbb{R}$ and $\varepsilon \in \{-1, +1\}$.
$$
f(x) = \begin{cases} a_1x + b_1, & \text{for } x \le 1 \\ a_2x + b_2, & \text{for } x \gt 1 \end{cases}
$$
with $a_1, a_2, b_1, b_2 \in \mathbb{R}$, can be represented as
$$
f(x) = m_1x + n_1 + \varepsilon|m_2x + n_2|, \text{ for } x \in \mathbb{R},
$$
where $m_1, m_2, n_1, n_2 \in \mathbb{R}$ and $\varepsilon \in \{-1, +1\}$.