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**bounded**function for which [itex]\sup\limits_{x\in [a,b]} f(x) \neq \max\limits_{x\in [a,b]} f(x)[/itex].

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lavinia

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boundedfunction for which [itex]\sup\limits_{x\in [a,b]} f(x) \neq \max\limits_{x\in [a,b]} f(x)[/itex].

f(x) = x except at the end points where it equals (a+b)/2

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[tex]

f(x) = \sum_{n = 0}^\infty \frac{n}{n+1} \chi_{\left[ \left. \frac{n}{n+1},\frac{n+1}{n+2} \right) \right.}(x).

[/tex]

Then [itex]f(x) \leq 1[/itex] for all [itex]x[/itex], [itex]\sup\limits_{x\in[0,1]} f(x) = 1[/itex], but [itex]f(x) \neq 1[/itex] for all [itex]x\in [0,1][/itex]. And, if I'm not mistaken, [itex]f[/itex] is still Riemann integrable, since there are only countably many points at which it is discontinuous. Can someone confirm if this is actually true?

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This is much easier than my example. Thanks!f(x) = x except at the end points where it equals (a+b)/2

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