Never read without a pencil in your hand. Let $\varepsilon>0$ and $\delta>0$ such that $$\mathop{\rm diam}f(E)<\varepsilon$$ for all $E\subset X$ with $\mathop{\rm diam}E<\delta$. Know your definitions in-and-out.. you should be able to write a definition out and its negation if need be. Mathematical Analysis Chapter 1 Part A. Linearity . They won't be easy. Hence, if any of the open intervals $(a_i,b_i)$ intersect $(x-\delta_1,x+\delta_2)$, then both $a_i$ and $b_i$ must be in $(x-\delta_1,x+\delta_2)$. If $\mathbf f(x)=\bigl(f_1(x),\ldots,f_k(x)\bigr)$ is a continuous map from a closed set $E$ in $\mathbf R$ into $\mathbf R^k$, then each of the component functions $f_n$ are continuous functions on $E$ by Theorem 4.10(a). Do lots and lots of problems. (By analambanomenos) Suppose $f$ is a uniformly continuous function from the metric space $X$ to the metric space $Y$. I would suggest (you probably meant this) not only watch them write the definitions and proofs, but write them down yourself. If $b_i=\infty$ define $g_i$ on $[a_i,\infty)$ to take the constant value $f(a_i)$. He does a great job of getting across how to think about definitions/results. Similarly, if $x+\delta\notin E$, we can replace $x+\delta$ with some $x+\delta_2=a_j\in E$. Thanks for the comments! Since $f$ and $g$ are continuous away from the origin, their restrictions to any line which doesn’t intersect the origin is also continuous. take a point set topology class and now undergrad analysis makes perfect sense, It's definitely helpful to see why a lot of the things about metric spaces that seemed "easy" to prove were easy because they hold more generally (e.g. (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. If $\delta_1=0$, so that $x=b_i$ for some $i$, then by the linearity of $g_i$ we can increase $\delta_1$ by an amount small enough so that $\big|g(x)-g(y)\big|= Let $p,q\in X$ such that $d_X(p,q)<\delta$. It is clear that $g$ is continuous at any $x\in(a_i,b_i)$, so suppose $x\in E$ and let $\varepsilon>0$. If you still have more free time, do the difficulty 5 ones. I’m of the opinion that analysis is basically trivial topology. Out of curiosity, how is Kolmogorov's real analysis book comaritively? Previous Post Solution to Linear Algebra Hoffman & Kunze Chapter 9.2. Actively write down the definitions, your thoughts on them, sketches of proofs, and so on. I am no longer in mathematics but I kept my Real Analysis notebook for almost sentimental reasons. Try to think of generalizations, or try coming up with examples for why a certain condition is actually necessary (or not). Let $g(x)=f(x)$ for $x\in E$ and $g(x)=g_i(x)$ for $x\in(a_i,b_i)$. If we let $E = X$, then Unless your class is using this book, don't go through it on your own. 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). Whenever the author says something is obvious, you should be able to explicitly prove it. By theorem 4.6 and the equality of $f$ and $g$ on $E$ \[ f(p) = \lim_{n \to \infty} f(p_n) = \lim_{n \to \infty} g(p_n) = g(p) \] so that $f$ and $g$ agree on all of $X$. If $\Gamma(E)$ is compact, let $V$ be a closed subset of $Y$. By Exercise 13, $f$ can be extended to a continuous function $\bar f$ on $\bar E$ whose range is also compact. The projection $ \pi : X \times Y \to X$ is continuous, so $ f^{-1}(V) = \pi(V^\prime) $ is compact, hence closed (since $X$ is a metric space and therefore Hausdorff). Solution to Principles of Mathematical Analysis Chapter 4 Part A, Solution to Principles of Mathematical Analysis Chapter 3 Part C, Solution to Principles of Mathematical Analysis Chapter 4 Part B, Solution to Principles of Mathematical Analysis Chapter 10, Solution to Principles of Mathematical Analysis Chapter 9 Part C, Solution to Principles of Mathematical Analysis Chapter 9 Part B, Solution to Principles of Mathematical Analysis Chapter 9 Part A, Solution to Principles of Mathematical Analysis Chapter 8 Part C, Solution to Principles of Mathematical Analysis Chapter 8 Part B, Solution to Principles of Mathematical Analysis Chapter 8 Part A, Solution to Principles of Mathematical Analysis Chapter 7 Part C, Solution to Principles of Mathematical Analysis Chapter 7 Part B, Solution to Principles of Mathematical Analysis Chapter 7 Part A, Solution to Linear Algebra Hoffman & Kunze Chapter 4.3.

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