# Continuity of $argmax$ of a strictly concave function

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.

• Such expression under argmax is strictly concave, not strictly convex. – szw1710 Oct 7 '17 at 23:10
• Oh, gosh, a typo, I'm correcting it right away! – Jules Oct 7 '17 at 23:17

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 $y$.