Young Functions: Properties & Relations

[fderingoz]

"In his studies on Fourier Series, W.H.Young has analyzed certain convex functions $\Phi$:IR$\rightarrow$$\bar{IR}$$^{+}$ which satisfy the conditions : $\Phi$(-x)=$\Phi$(x), $\Phi$(0)=0, and lim$_{x\rightarrow\infty}$$\Phi$(x)=+$\infty$. Then $\Phi$ is called a Young function.

Several interesting nontrivial properties and ordering relations can be analyzed if a Young function $\Phi$:IR$\rightarrow$IR$^{+}$ 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 $\Phi$:IR$\rightarrow$IR$^{+}$ is continuous-?

What is it that i can not see ?

 

[fresh_42]

Here's a paper which deals with it:
http://leonard.perso.math.cnrs.fr/papers/Leonard-Orlicz spaces.pdf