Monotonicity in argmin function

Question

I have the following equation $$f(a)=\text{argmin}_{t\in\mathbb{R}} \sum_{i=1}^n\frac{1}{2}(1-a)(b_i-t)^2+a| b_i-t |, a\in[0,1]$$

How does one prove that the function is monotone in a?

Answer 1

Suppose $a = 1$. Suppose further that $n = 2$ with $b_1 \neq b_2$. Then, any $t \in [b_1, b_2]$ is a minimizer. In other words, the minimizer is not unique. So $f(1)$ is a set. I doubt this is what you intended.