Complaint with Pugh’s real analysis textbook: He doesn’t even define the limit of a function properly?!
It’s implicitly defined together with the definition of continuity where ∀ϵ>0∃δ>0|x−x0|<δ⟹|f(x)−f(x0)|<ϵ, but in Chapter 3 when defining differentiability he implicitly switches the condition to 0<|x−x0|<δ without even mentioning it (nor the requirement that x0 now needs to be an accumulation point!) While Pugh has its own benefits, coming from Terry Tao’s analysis textbook background, this is absurd!
(though to be fair Terry Tao has the exact same issue in Book 2, where his definition of function continuity via limit in metric space precedes that of defining limit in general … the only redeeming factor is that it’s defined rigorously in Book 1, in the limited context of R)
*sigh* I guess we’re still pretty far from reaching the Pareto Frontier of textbook quality, at least in real analysis.
… Speaking of Pareto Frontiers, would anyone say there is such a textbook that is close to that frontier, at least in a different subject? Would love to read one of those.
Maybe you should email Pugh with the feedback? (I audited his honors analysis course in fall 2017; he seemed nice.)
As far as the frontier of analysis textbooks goes, I really like how Schröder Mathematical Analysis manages to be both rigorous and friendly: the early chapters patiently explain standard proof techniques (like the add-and-subtract triangle inequality gambit) to the novice who hasn’t seen them before, but the punishing details of the subject are in no way simplified. (One wonders if the subtitle “A Concise Introduction” was intended ironically.)
Complaint with Pugh’s real analysis textbook: He doesn’t even define the limit of a function properly?!
It’s implicitly defined together with the definition of continuity where ∀ϵ>0∃δ>0|x−x0|<δ⟹|f(x)−f(x0)|<ϵ, but in Chapter 3 when defining differentiability he implicitly switches the condition to 0<|x−x0|<δ without even mentioning it (nor the requirement that x0 now needs to be an accumulation point!) While Pugh has its own benefits, coming from Terry Tao’s analysis textbook background, this is absurd!
(though to be fair Terry Tao has the exact same issue in Book 2, where his definition of function continuity via limit in metric space precedes that of defining limit in general … the only redeeming factor is that it’s defined rigorously in Book 1, in the limited context of R)
*sigh* I guess we’re still pretty far from reaching the Pareto Frontier of textbook quality, at least in real analysis.
… Speaking of Pareto Frontiers, would anyone say there is such a textbook that is close to that frontier, at least in a different subject? Would love to read one of those.
Maybe you should email Pugh with the feedback? (I audited his honors analysis course in fall 2017; he seemed nice.)
As far as the frontier of analysis textbooks goes, I really like how Schröder Mathematical Analysis manages to be both rigorous and friendly: the early chapters patiently explain standard proof techniques (like the add-and-subtract triangle inequality gambit) to the novice who hasn’t seen them before, but the punishing details of the subject are in no way simplified. (One wonders if the subtitle “A Concise Introduction” was intended ironically.)