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Here's proof of Royden's real analysis: enter image description here

My question is do we real have bounded interval $[f(a),f(b)]$. Because if we let $f(a)=-\infty$ and $f(b)=\infty$, so is this proof wrong?

By the way, I prefer to prove this in this way:How to show that a set of discontinuous points of an increasing function is at most countable .

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DuFong
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    Royden assumes that $f$ is increasing on the closed interval $[a,b]$ in the second sentence of the proof. – Daniel Fischer May 11 '16 at 18:50
  • @DanielFischer But it still cannot imply interval $[f(a),f(b)]$ bounded, right? – DuFong May 11 '16 at 18:51
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    If $f$ is assumed real-valued rather than extended-real-valued, then ${f(a), f(b)} \subset \mathbb{R}$, so the interval is bounded. – Daniel Fischer May 11 '16 at 18:53
  • You are right. Then how about the extended-real-valued case? – DuFong May 11 '16 at 18:56
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    Slight modification. Assuming that $f$ attains both infinities, let $\alpha = \sup { x : f(x) = -\infty}$ and $\beta = \inf { x : f(x) = +\infty}$. Apply the given proof to $f\lvert_{(\alpha,\beta)}$, and note that all discontinuities of $f$ lie in $[\alpha,\beta]$. – Daniel Fischer May 11 '16 at 18:59

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