I'm supposed to invent a continuous function f: R into R such that its improper integral is zero, but which is unbounded as x approaches negative infinity and x approaches infinity.
I have no clue how to do this problem. Could somebody help me? Thanks
How about such that for all , is defined on every by and on ?
On each interval , the curve of the function has the shape of a triangle which has one vertex at , one at and one at . As approaches , the height of the triangle does the same (see the second vertex) so the function is not bounded. It can be checked that this function is continuous on . As, for , (it's the area of a triangle which height is and which base is ), we get that and the function is integrable.
As we want , we define for , and that's it.
If you write out the formulas for this construction, you get exactly what flyingsquirrel wrote.