Young functions

  1. "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. "(rao-ren 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 ?
     
  2. jcsd
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