Terence Tao says the following in the preface to his book Analysis I:
With regard to examinations for a course based on this text, I would recommend either an open-book, open-notes examination with problems similar to the exercises given in the text (but perhaps shorter, with no unusual trickery involved), or else a take home examination that involves problems comparable to the more intricate exercises in the text. The subject matter is too vast to force the students to memorize the definitions and theorems, so I would not recommend a closed-book examination, or an examination based on regurgitating extracts from the book. (Indeed, in my own examinations I gave a supplemental sheet listing the key definitions and theorems which were relevant to the examination problems.)
Based on Tao's comments, we can separate the most common types of examination into the following categories:
- Closed-book/closed-notes examination without supplemental sheet.
- Closed-book/closed-notes examination with supplemental sheet.
- Open-book/open-notes examination.
- Home examination.
Based on your experience (or on sources on the subject that you know), in the context of real analysis examination:
(a) What is the best category?
(b) What are the pros and cons?
(c) How similar should the exam questions be to the homework questions?
If you use a different category from those listed above, feel free to comment.
Addendum: I'm talking about an introductory course that covers the basics of natural and real numbers, convergence of sequences and series, limits of functions, continuity, derivatives, integrals, sequences and series of functions.