Continuity and close map . [Metric Space.]

Question

The question is - Show that the function $f\colon\mathbb{R}\times\mathbb{R}\to \mathbb{R}$ defined by $f(x,y)=x$ is continuos but it is not closed ..I don't know how to prove this, I am approaching in this way, I am analyzing this question and found that this function is onto but not one one. i have to show that it is not a closed map , i.e it does not send closed sets onto some closed sets but i take some examples like a unit circle then f(unit circle) comes out to be [-1,1] which is closed in R , i take many other examples as well but cannot be able to contradict this ,, also how to claim it is continuos map , iam proceeding like this let i need to check continuity at the point (1,2)- let epsilon > 0 is fixed now consider d(f(x,y)-f(1,2)) =d(x,1)=mod(x-1) , so i need to find a delta>0 such that if (x-1)^2+(y-2)^2 < delta then mod(x-1)<=mod(1+ delta-1)=delta

Answer 1

HINT: Consider $g(x)=1/x$ and projection of its graph.