Let E be a normed vector space. Let be defined by , the derivative of . Show that the definition of a continuous function fails at zero.
Forget , we can do this for . So ,if were continuous at then we should be able to find some such that , right? But, for example if you claimed to me that you have found a which works I can show you that you're wrong. To see this we know by the Archimedean principle I can find some such that and , right? So, consider then clearly but , and and so the you claimed work didn't, and so you can't find any which works. So, it isn't continuous. Make sense?