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.
2026-04-12 04:44:52.1775969092
A bounded function having I.V.P. but not Riemann Integrable.
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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.
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.