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Young functions 
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#1
Feb2312, 09:05 AM

P: 12

"In his studies on Fourier Series, W.H.Young has analyzed certain convex functions [itex]\Phi[/itex]:IR[itex]\rightarrow[/itex][itex]\bar{IR}[/itex][itex]^{+}[/itex] which satisfy the conditions : [itex]\Phi[/itex](x)=[itex]\Phi[/itex](x), [itex]\Phi[/itex](0)=0, and lim[itex]_{x\rightarrow\infty}[/itex][itex]\Phi[/itex](x)=+[itex]\infty[/itex]. Then [itex]\Phi[/itex] is called a Young function.
Several interesting nontrivial properties and ordering relations can be analyzed if a Young function [itex]\Phi[/itex]:IR[itex]\rightarrow[/itex]IR[itex]^{+}[/itex] is continuous. "(raoren theory of orlicz spaces 1991) I think we can say from the definition of young function : Young functions are convex functions on IR and IR is a open convex set and we know also that if a funtion is convex on an open convex set then this function is continuous on that open set, So young functions are continuous. Why the authors needs to write second paragraph,i.e. Several interesting nontrivial properties and ordering relations can be analyzed if a Young function [itex]\Phi[/itex]:IR[itex]\rightarrow[/itex]IR[itex]^{+}[/itex] is continuous? What is it that i can not see ? 


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