let f:X -> (0,inf) be a continuous function on a compact space X. show that there exits an ε >0 such that f(x)>=ε for all x element of X
Question
Let f: X -> (0,inf) be a continuous function on a compact space X. Show that there exists an such that for all .
Solution
To prove this, we will use the properties of continuous functions on compact spaces.
-
Since f is continuous on the compact space X, the image of X under f, denoted as f(X), is also compact. This is a well-known property of continuous functions.
-
The set f(X) is a subset of (0, ∞), which is an ordered set. Therefore, f(X) has a least element, say ε. This is because every non-empty subset of the real numbers that is bounded below has an infimum (greatest lower bound).
-
Since ε is the least element of f(X), we have f(x) ≥ ε for all x in X. This is by the definition of the least element.
Therefore, there exists an ε > 0 such that f(x) ≥ ε for all x in X.
Similar Questions
Let f, g : R → R be given functions. Suppose that f and g are continuous at c ∈ R.Prove that the functionl(x) := inf{f (x), g(x)}, x ∈ R,is continuous at c.
Suppose that S and T are nonempty subsets of R which are bounded below, and thatS ⊆ T . Prove that inf S ≥ inf T
Consider the function f : R → R defined by f (x)=1 if x∈ Q, f (x)=0 if x∈ R/Q, where is f continuous? be sure to prove your assertion
Let a, b ∈ R. Suppose that for every ε > 0, we have a < b + 3ε. Prove that a ≤ b
Let [a, b] be a bounded closed interval, f ∈ R[a, b] and M > 0. Suppose that |f (x)| ≤ Mfor all x ∈ [a, b]. Prove thatZ baf ≤ M (b − a)
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.