I don't know the meaning of these: 1) [tex]sup_{B_\delta}|f(x,y)| [/tex] Where [itex]B_\delta [/itex] is the ball of radius [itex]\delta[/itex]. 2) [tex]\int \int _{R^2 \B _{\delta} } f(xy)dxdy[/tex] I don't know what is [itex]R^2 [/itex]\B[itex] _{\delta} [/itex] Please read my latex because the symbol really don't show correctly.
1) The supremum of {|f(x,y)|} where (x,y) ranges over the ball centered at 0 of radius delta: |(x,y)|=sqrt(x^2+y^2)<delta. 2) the plane R^2 without the ball centered at 0 of radius delta, i.e. \ (in Latex: "\backslash") means 'complement' or 'set difference'. So it consists of pairs (x,y) of real numbers such that |(x,y)|=sqrt(x^2+y^2)>=delta.
Thanks for you reply, so for 1) Is the upper bound of |f(x,y)| in the ball. 2) Is the whole 2D plane minus the circle center at 0 with radius [itex]\delta[/itex]
No. There is no such thing as "the" upper bound of a set of numbers. If a set has an upper bound, then it has an infinite number of upper bounds. This is the least upper bound- the smallest number in the set of all upper bounds.