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Chapter 5: Problem 4
Show that if \(f\) and \(g\) are positive increasing functions on an interval \(I\), then their product \(f g\) is increasing on \(I\).
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Examine the mapping of open [respectively, closed) intervals under the functions \(g(x):=\) \(1 /\left(x^{2}+1\right)\) and \(h(x):=x^{3}\) for \(x \in \mathbb{R}\).
Let \(f, g\) be increasing on an interval \(I \subseteq \mathbb{R}\) and let
\(f(x)>g(x)\) for all \(x \in I .\) If \(y \in f(I) \cap g(I)\), show that
\(f^{-1}(y)
Let \(I:=[a, b]\), let \(f: I \rightarrow \mathbb{R}\) be continuous on \(I\), and assume that \(f(a)<0, f(b)>0\). Let \(W:=\\{x \in I: f(x)<0\\}\). and let \(w:=\sup W\). Prove that \(f(w)=0\). (This provides an alternative proof of Theorem \(5.3 .5 .)\)
If \(f\) is uniformly continuous on \(A \subseteq \mathbb{R}\), and \(|f(x)| \geq k>0\) for all \(x \in A\), show that \(1 / f\) is uniformly continuous on \(A\).
If \(g(x):=\sqrt{x}\) for \(x \in[0,1]\), show that there does not exist a constant \(K\) such that \(|g(x)| \leq K|x|\) for all \(x \in[0,1]\). Conclude that the uniformiy continuous \(g\) is not a Lipschitz function on \([0,1]\).
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