I'm examining the final paragraph on page 75 from, Oden - Applied Functional Analysis 2nd ed., and am hoping for clarity.

A statement is made that for a function such as using the extended real line where the value at x = 0 is , is continuous. The author then goes on to say that this continuity is not true for at x=0.

I am looking for illumination as to why this is the case. Thank you.

I am not sure if this may be of help but the section is "Functions with values in Bar-R (Extended Real Line)." Previous discussions include point-wise sup/inf.

Chris