Question

Let K ⊂ be a compact and f : K → [0, [infinity][ a continuous function. Show that the set Γ = (x, y) ∈ K × is a compact subset of +1 (we identify +1 with × ).

Answer 1

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

Step-by-step explanation:

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

To show that Γ is closed, we can use the fact that f(x) is continuous. This means that for any sequence {(x_n, y_n)} in Γ that converges to a point (x, y), we have f(x_n) converging to f(x), and since f(x) is in [0, ∞), it implies that y_n converges to y which shows that (x, y) is in Γ.

To show that Γ is bounded, we can use the fact that K is a compact set. Since Γ can be expressed as the set union of all vertical lines x = c, 0 ≤ c ≤ f(x), and each vertical line segment is contained in the compact set K, Γ is contained in the set K × [0, ∞) = K × +1, which is a compact set.