Loss optimization of $max$ function

Question

I have to find the function $f(x)$ that minimizes the expression $$ L(y, f(x)) = \left\{\begin{array}{ll} \max(0, 1 - yf(x)), & yf(x) \geq 0 \\ 1 - kyf(x), & otherwise\end{array}\right.$$

where $y \in \mathbb{R}$ and $k \geq 1$.

I know that I may have to take a look at the different intervals, but I don't seem to find a start.

Answer 1

By simple inspection, the minimum value of $L(y,f(x))$ is zero. Thus, for any $y \in \mathbb{R}-\{0\}$, the constant functions $f(x) = c$ such that $cy >1$ are optimal solutions.

Edit:

Actually, all functions $f(x)$ such that $$\min \limits_{x} ~~ f(x) \geq c,$$ are also solutions.