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how would you show that $$f(x) = argmax_{y\in\mathbb{R}}\{ay+bx+c-\left|\left|y-x\right|\right|^2\}$$ is continuous? It is well defined since the expression under argmax is strictly concave and thus is has just one maximum. But why the continuity?

I feel it is very simple but I just can't figure it out. I kindly ask for a hint.

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  • $\begingroup$ Such expression under argmax is strictly concave, not strictly convex. $\endgroup$ – szw1710 Oct 7 '17 at 23:10
  • $\begingroup$ Oh, gosh, a typo, I'm correcting it right away! $\endgroup$ – Jules Oct 7 '17 at 23:17
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Firstly, the expression inside argmax is not convex, it is concave. Secondly, you can find an analytical solution for $f(x)$ by expanding the quadratic term and complete the square involving $y$.

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