# Integrable function

by reza
Tags: function, integrable
 P: 26 how can we prove that if F(function) is integrable [a,b] then f must be bounded on [a,b]
 Sci Advisor HW Helper PF Gold P: 4,771 By definition, a function in integrable if the lower integral equals the upper integral. What happens to the upper integral if a function is not bounded, say, above?
P: 1,295
 By definition, a function in integrable if the lower integral equals the upper integral. What happens to the upper integral if a function is not bounded, say, above?
This, the Darboux definition, is equivalent to the Reimann-Stieltjes definition, which the OP may prefer to work with for "class" reasons.

The easiest way to see this result is by the contrapositive. If f is unbounded, the integral of f does not exist.

 Sci Advisor P: 6,070 Integrable function Something is missing here. x-1/2 is integrable between 0 and 1, but it is not bounded.
 P: 1,295 Integration is defined only for closed intervals, an improper integral is an extension of this.
 P: 26 can you give me a mthemathica proof