It is known that and we say that is differentiable at if this limit exists. But if this limit is or as , can we still say that is differentiable at , or more precisely, is this limit still considered to exist? Thanks.
Also; you are now changing the question, by default in real analysis we are talking about the real, if now we are talking about some version of the extended reals we have a different question.
Thank you for your reply. I'm reading Rudin's "Principles of Mathematical Analysis", this book introduces infinite limits(P98) just before the definition of derivatives(P103), so I naturally have the question if infinite limit is regarded as "exists" or not. According to your reply, it is not, so I concluded that we have to confine ourselves to traditional real set in dealing with limits of functions in the definition of derivatives, but I'm not sure, that's my second post. I have refered to other books such as "Mathematical Analysis" written by Apostol, but all these books do not explain explicitly whether infinite limit is allowed. Could you please cite some textbooks stating explicitly that, in the definition of derivatives, infinite limit means the limit does not exist? Thanks!
ps: I have not study real analysis yet. I plan to study it after I finish baby Rudin, and to use Rudin's "Real and Complex Analysis" as textbook, is this book good for self-study?
I finally find an explict answer from wiki. In brief, the limit of the difference quotient exists if the quotient can be "continuouslized" by supplementing a value at zero. Since the codomain of continuous function is only the real set, can not be used as a value for the difference quotient. So, infinite limit means being not differentiable. In addition, the wiki webpage gives an explicit statement that functions with vertical tangent at some point, that is, the limit is , is not differentiable there.
Thank you all.
Although my question is solved, I'd like to say something about the existence of infinite limit. when I studied advanced calculus, I often met notations such as , or . At first I just cannot understand what the symbol " " actually is, of course it is not a real number, but it occurs frequently in the textbook, with some strange properties such as . Sometimes I have to write by hand but says "Oh, it diverges" in heart. This antithesis lasts for years until I studies the general topology, especially the notion of compactification. There, " " together with real set constutute a compact space, is just a common element which is nothing different from real numbers, and or which means divergence in advanced calculus IS indeed a convergence in topology. In topology the convergence means there is always a -neighborhood of the variable such that the values can all fall into any -neighborhood. To be more precise in limits of functions (assuming defined everywhere), converges to at iff for any neighborhood of (here I use basis element instead), there is a neighborhood of excluding , that is, , such that whenever . Since is an order topology, the neighborhood of is just where is any real, so the traditional convergence is of course convergence, but for , that is, for any real , no matter how large it may be ,there is a in a -neighborhood of such that . But look at what means? it just means , a neighborhood of ! so according to the aforementioned definition of convergence in topology, DOES CONVERGE to , NOT diverge. So it is not an abuse of the word "limit" or "converge", nor is the limit inexistent. As for my question, although it is often confused if is allowed in a notion, the codomain of a function is definite, we are always working with functions in , i.e., the codomain is , not including ; there is no confusion regarding this. So in the definition of derivative, using the "continuouslization" of the difference quotient, we can conclude that infinite limit means the derivative does not exist, although the function limit does exist -- = . As a result, textbooks should not only says the limit exists, but also should add a note that the limit not be in defining derivatives. To sum up, the question is not whether the limit exist, but whether we should confine or restrict ourselves to real set only when understanding this concept.
Since we have learned topology, why not hold a unified point of view, although sometimes at the cost of confusion whether is included or not.