I would really appreciate if anyone can give me a formal citation for this. I know this is a standard problem for which I also have a proof. However this is requried as a part of a more complicated problem and it would be nicer for the writing if I can just simply give a citation.
Just say it follows from compactness and continuity that....
If you want, put the proof as a lemma in the appendix.
For the same question see
http://math.stackexchange.com/questions/20300/infimum-is-a-continuous-function-compact-set/1378023#1378023
and discussion there. It turns out that it is an interesting question which may related to the max -min theory.
Now we will outline a proof.
Perhaps, we can suppose that X and Y are metric spaces (in particular subset of real line R) and that f(x,y) is real-valued function in x ,y. For a simple model we can take X=R and Y=[a,b] closed interval in R.
Set I(x) = inf_{y\in Y} f(x,y). Fix a point x_0 in X .
Let us prove that I(x) is continuous at x_0.
Consider a sequence x_n in X such that x_n converges to x_0.
Let us prove that I(x_n) converges to I(x_0).
Since Y is compact there is a sequence o (y_n) and y_0 in Y such that I(x_n) = f(x_n,y_n) and a_0:= I(x_0= f(x_0,y_0). Since Y is compact there is a subsequence of y_n which converges to y^0 in Y. By abusing notation we can suppose that (x_n,y_n) converges to (x_0, y^0). By continuity of f in x ,y we conclude that f(x_n,y_n) converges to f(x_0, y^0):=b_0. By definition of I(x_n), f(x_n,y_n)
I would recommend looking at the following paper by Clarke (1975): www.ams.org/tran/1975-205-00/.../S0002-9947-1975-0367131-6.pdf . This paper gives some very general results on functions of the form f(x) = max {g (x,u) : u \in U} (such as when they are continuous or differentiable). It has 1099 citations, according to Google. And I believe it is a well-known paper in optimization circles.
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