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
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Solution 1

To prove this, we will use the properties of continuous functions on compact spaces.

  1. 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.

  2. The set f(X) is a subset of (0, ∞), which is an Knowee AI StudyGPT is a powerful AI-powered study tool designed to help you to solve study prob

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