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I am searching an example of a function $f$ on $[a,b]$ such that $f$ is a bounded function having intermediate value property but is not Riemann Integrable on $[a,b].$ Please give me such type simplest example which can be easily visualized. Thanks in advance.

Martin Sleziak
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neelkanth
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    The derivative of anything has the intermediate value property. Accordingly, you want the derivative of Volterra's function (see wiki). – Ian Dec 11 '15 at 12:04
  • please if possible write the simplest function.. – neelkanth Dec 11 '15 at 12:04
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    There will be no simple construction, because you have to put oscillatory discontinuities (like the one in sin (1/x)) at every point on a set of positive measure. In a sense Volterra's construction is the most obvious way to do this. – Ian Dec 11 '15 at 12:06
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    Any of the examples given as answers in this question have the required property. (You can find there also links to many other posts with similar functions.) – Martin Sleziak Dec 17 '15 at 08:12

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For more than you might care to know about Darboux functions (also known as "functions with the IVP") see this survey article.

Bruckner, A. M.; Ceder, J. G. Darboux continuity. Jber. Deutsch. Math.-Verein. 67 1964/1965 Abt. 1, 93–117.

Your request, however, for "easily visualized" is a bit demanding. Ian's suggestion of the Volterra example will require you to know about Cantor sets of positive measure. If you can visualize those things then, indeed you can "picture" what such a function looks like.

The easiest example to my taste can be found in the survey article: it uses Hamel bases. If you have the right background that is a nice example of an everywhere discontinuous Darboux function. (Not bounded but you could chop into a bounded function.) If you are sufficiently crazy (like most mathematicians) then Hamel bases can be visualized well enough.

This is an interesting topic and I rather hope that your question opens some doors for you and leads you somewhere new.

B. S. Thomson
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