It is clear that $g$ is continuous at any $x\in(a_i,b_i)$, so suppose $x\in E$ and let $\varepsilon>0$. Chapter 3 Numerical Sequences and Series. That's a great point. Never read without a pencil in your hand. Maybe skip some of Chapter 1. Get Rudin and some other book with pictures/drawings (I liked Abbott's and Pugh's). I like it. Actively write down the definitions, your thoughts on them, sketches of proofs, and so on. Out of curiosity, since your advice does stand out a lot, how would you estimate your 'level of doing maths' or how serious you take mathematics? Its more comparable to rudins real and complex analysis book. (1) be confused at my professor writing definitions on the board, (2) grind myself to the bone working out problems and never actually understanding anything, (4) take a point set topology class and now undergrad analysis makes perfect sense, Analysis didn't click for me until I took my 3rd semester when we proved everything in Rn. (By analambanomenos) Suppose $f$ is a uniformly continuous function from the metric space $X$ to the metric space $Y$. \big|g_i(x)-g_i(y)\big|<\varepsilon$ for $y\in(x-\delta_1,x)$. Do lots and lots of problems. That's definitely also a possibility. My (4) was “fall asleep in topology every day and learn it minimally – become a theoretical computer scientist” but the rest of my list was the same. Why are you assuming the OP is a complete novice? I honestly don’t think I could have counted the number of times I had just learned something in analysis and then had it mentioned in topology as a trivial consequence of such and such theorem of Urysohn or Alexandroff or whatever. (By analambanomenos) Note that both $f$ and $g$ are equal to 0 on the $x$ and $y$ axes. (By analambanomenos) The quickest solution uses Exercise 13 below. They were a thing of beauty. Maybe skip some of Chapter 1. If $p,q\in E$, then $d_X(p,q)\le\mathop{\rm diam}E<\delta$, so $d_Y\bigl(f(p),f(q)\bigr)<\varepsilon$. Even if you can go through each proof line by line (which I doubt you'll be able to, but that's an aside), without a teacher to explain why a particular proof is important or why the book is laid out in the manner in which it is, you're going to go through an awful lot of effort which would be better spent on a more intuitive book (read: one with more detail between theorems and more detail in proving theorems). Every now and then, when you feel like going through a single problem for hours, do a difficulty 4 (you should do this, it is a part of mathematics to spend several hours bashing your head against the wall until you solve an exercise). Chapter 4 Continuity Part A: Exercise 1 - Exercise 9 Part B: Exercise 10 - Exercise 18 Part C: Exercise 19 - Exercise 26 Exercise 1 (By ghostofgarborg) No. It's called "baby" because he has a more advanced version as well). Let $f(x)=x^{-1}$ for $x\in(0,1)$. Since $f$ and $g$ are continuous away from the origin, their restrictions to any line which doesn’t intersect the origin is also continuous. Let $E\subset X$ with $\mathop{\rm diam}E<\delta$. I should add that spivak's calculus is a very good first primer in formal proofs if cracking open rudin seems intimidating. Walter Rudin (May 2, 1921 – May 20, 2010) was an Austrian-American mathematician and professor of Mathematics at the University of Wisconsin–Madison.. Every now and then I see people asking questions about Rudin's book in here and so I'd like to share how I approached it (succesfully) two years ago when going through my first real analysis. I have had analysis that taught a lot of the material in book one so I want to self-study book two. (By ghostofgarborg) Let $U \subset f(X)$ be open. Do not just copy these solutions. By the construction of the $g_i$, we must have $\big|g(x)-g(y)\big|<\varepsilon$ for $y\in(x-\delta_1,x+\delta_2)$. Download George Bergman's notes about Rudin's exercises. While I can understand wanting to give your child a strong foundation in math, I'd probably recommend something with colorful pictures for an infant. Tags: Baby Rudin. You should attempt to prove the non-intimidating theorems yourself first before reading their proofs. Completely different grade. Definitely. Good luck. For $y\neq 0$, $g(ky^3,y)=k/\bigl((k^3+1)y\bigr)\rightarrow\infty$ as $y\rightarrow 0$, so $g$ is unbounded in every neighborhood of $(0,0)$. Continue Reading. Since the value of $f$ along the parabola $(ky^2,y)$ drops from $k/(k^2+1)$ to 0 at $y=0$, $f$ is not continuous at $(0,0)$. Unless your class is using this book, don't go through it on your own. For example, if you are in college, what kind of college are you in and how would you compare yourself to your classmates? (4) was close to home. It's a better reference than a learning tool. Rudin writes as if his audience is already pretty familiar with math, which shouldn't be the case for someone learning basic analysis.