Question

Let K ⊂ Rd be a compact and f : K → [0,[infinity][ a continuous function. Show that the set Γ = (x, y) ∈ K × R

is a compact subset of Rd+1 (we identify Rd+1 with Rd × R).

attached the exercise in french

Answer 1

To show that the set Γ = 0 ⩽ y ⩽ f(x) is a compact subset of Rd+1, we need to show that it is closed and bounded.

Step-by-step explanation:

To show that the set Γ = 0 ⩽ y ⩽ f(x) is a compact subset of Rd+1, we need to show that it is closed and bounded.

1. Closed: Since f(x) is continuous on a compact set K, it is also uniformly continuous. This means that for every ε > 0, there exists δ > 0 such that |f(x) - f(y)| < ε whenever |x - y| < δ. This implies that for any sequence {x_n} converging to x ∈ K, the sequence {f(x_n)} converges to f(x). So, y = f(x) is continuous and hence closed.

2. Bounded: Since K is compact, it is bounded. And since f(x) is continuous on a bounded set, it is also bounded. Therefore, the set Γ = 0 ⩽ y ⩽ f(x) is bounded.

Thus, Γ is a closed and bounded set in Rd+1, which means it is a compact subset of Rd+1.