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 exits an ε >0 such that f(x)>=ε for all x element of X
Solution
To prove this, we will use the properties of continuous functions on compact spaces.
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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.
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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).
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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.
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