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I know that the Riemann Integral is defined for every piecewise continouus curve.

But now suppose you´re asked the following:

you are given [tex]f(x,y)=\frac{xy^3}{(x^2+y^2)^2}[/tex] with additional Definition

f(0,0)=0. ( It´s a textbook problem :) )

Now surely f(x,y) is smooth everywhere except for the origin. So I tried to show that f(x,y) goes to 0 in the origin but it doesn´t goes to 1/4(easy to show by setting x=y because f(x,y)=1/4 for x=y).

So the book asks to proof that the function [tex]g(y)=\int_{0}^{1}f(x,y)dx [/tex] is well defined on the real line.

I would intuitively say no because of the discontinuity at the origin but otherwise the origin is a set of measure zero.

So how does that definition go ?

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# Exact Definition of Riemann Integral

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