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SUMMARY

The discussion focuses on the properties of continuous functions and bounded sets in real analysis. It establishes that if \( f: \mathbb{R} \to \mathbb{R} \) is continuous and \( B \) is a bounded subset of \( \mathbb{R} \), then the closure \( \text{cl}(B) \) is also bounded. Furthermore, it concludes that the image set \( f(B) \) must be a bounded subset of \( \mathbb{R} \). The discussion also raises the question of whether the image set \( g(B) \) is necessarily bounded when \( g \) is continuous on \( B \) but not on all of \( \mathbb{R} \.

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Suppose f: R->R is continuous on all of R and B is bounded subset of R.
a) show cl(B) is bounded set
b) show image set f(B) must be bounded subset of R
c) suppose g:B->R is defined & continuous on B but not necessarily on all of R - real #s, Must g(B) be bounded subset of R? (Prove or give counterexample)
 
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