3
votes
1answer
24 views

Find extrema of $f_a(x)=\vert x-a\vert^2$ on $\overline{B_1(0)}$

Let $\overline{B_1(0)}\subseteq\mathbb R^3$ be the closed unit-sphere and $a\in\mathbb R^3$. Find all extrema of the function $f_a(x)=\vert x-a\vert^2$ on $\overline{B_1(0)}$ depending on $a$. ...
2
votes
1answer
219 views

A sequence converges $\iff$ it's Cauchy. Proof of ($\Leftarrow$) (Abbott p 59 t2.6.4)

Lemma 2.6.3 $\implies (x_{n})$ is bounded. So use the Bolzano-Weierstrass Theorem to produce a convergent subsequence $(x_{n_{k}})$ . Set $x= \lim x_{n_{k}}.$ So $(x_{{n_{k}}}) \to x. \quad ...
2
votes
3answers
88 views

Intuition. Equivalence of Characterization of Limits and Continuity (Abbott p106 t4.2.3, p110 t4.3.2)

What are the intuitions of these equivalences? Not questioning about proofs or any rigour. I question both equivalences jointly because they look similar. And Are there any figures? ...
1
vote
2answers
72 views

if $g$ is continuous at $c$ and $g(c)\neq 0$, there exists an open interval containing $c$ on which $f(x)/g(x)$ is defined (Abbott p 113 q4.3.5)

Theorem 4.3.4.(iv) says that $f(x)/g(x)$ is continuous at $c$ if both $f$ and $g$ are, provided that the quotient is defined. Show that if $g$ is continuous at $c$ and $g(c)\neq 0$, then there exists ...
1
vote
1answer
40 views

Directional derivatives in any direction do not imply continuity?

I found an example where a function from R^2toR has directional derivatives at a point p at any direction however the function isn't continuous at p. I found this very weird because I thought that ...
2
votes
3answers
92 views

How do we arrive at the definite integral to find area approximated by a sum of rectangles?

The area enclosed by a one variable function from a to b can be approximated by $n$ rectangles$$A \approx \sum_{i=1}^{n} f(x_i)(x_i-x_{i-1})$$ and if we let $n \rightarrow \infty$ we get $$A = ...
3
votes
2answers
88 views

Show that $f(x,y)= \|x-y\|_2^2$ is differentiable

Problem: Show that $f: \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ with $f(x,y)=\|x-y\|_2^2$ is differentiable and compute its differential at every point in the domain of $f$Note: $\| \cdot ...
1
vote
1answer
57 views

Intuition behind a convergent subsequence of $\sin(n)$

$\let\eps\varepsilon$ I was looking through a Real Analysis exam paper one day and was stuck on a question; fortunately there is a solution provided which I will sketch below, but I have no intuition ...
0
votes
2answers
42 views

Limit Point of a Set

Definition. A point $x$ is a limit point of a set A if every $\varepsilon$-neighborhood of $x$ intersects the set A in some point other than $x$. I understand the definition in that $x$ is our limit ...
0
votes
0answers
28 views

Determine weaker hypotheses for Evaluation Fundamental Theorem of Calculus (Abbott p 202 T7.5.3)

(p 200 T7.5.1) If $f:[a,b] \to R$ is integrable and $F:[a,b] \to R$ satisfies $f(x)= f'(x) \; \forall x \in [a,b]$, then If $g$ is integrable on $[a,b]$, then $\int_a^b f = F(b) - F(a)$. ...
1
vote
2answers
107 views

What does Riemann-Stieltjes integral calculate when $\alpha(x) \neq x$?

When we get Riemann-Stieltjes integral becomes standard Riemann integral which calculates area under the curve. We have that $$ s(f,\alpha,P)=\sum_{k=1}^nm_k\Delta\alpha_k \ \text{ and }\ ...
4
votes
0answers
90 views

Visualize $f(b) - f(a)$ withOUT Mean Value Theorem (Stewart p 282 figure 4) [closed]

How can we visualize $\color{green}{f(b) - f(a)}$ withOUT the Mean Value Theorem or rewriting it as $\color{dodgerblue}{\dfrac{ f(b) - f(a) }{ b - a }} $ ? I'm trying to understand ...
2
votes
2answers
81 views

Modus operandi for proving Evaluation Fundamental Theorem of Calculus (Abbott p 200, Spivak p 272 T14.2)

1. How can we presage to use Mean Value Theorem to start the proof? 2. Mean Value Theorem engenders a point in an open interval. Shouldn't this be $x_i \in (t_{i - 1}, t_i) $? After ...
2
votes
0answers
84 views

Choices of epsilons in proof : $(b_n) \to b$ implies $\left\{\frac{1}{b_n}\right\} \to \frac{1}{b}$ (Abbott pp 47 T2.3.3.iv) [closed]

Original became long, ergo I ask anew. The trick is to look far enough out into the sequence $(b_n)$ so that the terms are closer to b than they are to 0. Consider the particular value $e_0 = |b|/2$. ...
1
vote
2answers
78 views

Is this a counterexample to “continuous function…can be drawn without lifting” ? (Abbott P111 exm4.3.6)

I'm au courant with http://math.stackexchange.com/a/288133 and http://math.stackexchange.com/a/422001. They're both Abbott P111 exm 4.3.6 which proves "a continuous function is sometimes described, ...
1
vote
1answer
50 views

How to presage Prove by Contrapositive, for Sequential Characterizations of Limit and Continuity? (Abbott pp 106 t4.2.3, 110 t4.3.2)

Dafinguzman answered consummately this question initially but it became too long. I want to question for different beliefs. 1. $(ii) \implies (i)$ in both Theorems 4.12 and 4.19 posit sequences ...
1
vote
1answer
57 views

Intuition and counterexamples for higher-order derivative test

In the higher-order test we keep differentiating a function till we find the n'th derivative (n being even) to be greater than or less than zero thereby identifying it as a minimum or maximum. My two ...
1
vote
0answers
58 views

Intuition for $\inf(AB) = \inf(A)\sup(B)$. Difference for sets and functions? (Abbott pp 199 q7.4.5)

1. What's the intuition for $\inf(AB) = \inf(A)\sup(B)$? Figure please? I know I must posit $A,B \subseteq R$ as bounded sets. If they're unbounded, $\sup$ doesn't exist. I believe $\inf(AB) = ...
1
vote
0answers
76 views

So we don't need to choose delta, epsilon, or $N \in \mathbb{N}$ in delta-epsilon or sequence convergence proofs?

(http://math.stackexchange.com/a/700667/85079) I would write the proof with all my bounds $\eta$ and then choose $\eta$ to make the conclusion match the arbitrary $\epsilon$. ...
1
vote
1answer
73 views

If $g \ge 0$ is continuous on $[a,b]$ and $g(x_0) > 0$ then $\int^{b}_a g > 0$ (Abbott pp 199 q7.4.4c)

True or False. If $g \ge 0$ is continuous on $[a,b]$ and $g(x_0) > 0$ for $\ge 1$ point $x_0 \in [a,b]$, then $\int^{b}_a g > 0.$ 1. Need determine if true or false. Ergo do we need ...
1
vote
3answers
93 views

If $\int^{b}_a f > 0$ then there is some interval and $\delta > 0$ on which $f(x) \ge \delta$ (Abbott pp 199 q7.4.4d)

True or False. If $\int^{b}_a f > 0$, then $\exists \; [c,d] \subseteq [a,b]$ and $\delta > 0$ such that $f(x) \ge \delta$ for all $x \in [c,d]$. 1. We need to determine if true or false. ...
2
votes
1answer
53 views

Proof. sup{ f(x) } - inf{ f(x) } $\ge$ sup{ |f(x)| } - inf{ |f(x)| } (Abbott pp 198 q7.4.1)

Let f be a bounded function on a set A, and set $S = \sup\{f(x) : x ∈ A\}, I = \inf \{f(x) : x ∈ A\},$ $S' = \sup\{|f(x)| : x ∈ A\}, I' = \inf \{|f(x)| : x ∈ A\}.$ Show that $S - I ≥ S' - I'$. ...
1
vote
0answers
18 views

Intuition. 3 Equivalences of Riemann integrability (Abbott pp 191 q7.2.4)

Not questioning about proofs. For this entire question, posit $f$ is a bounded function on $[a,b]$. ♪ f is integrable signifies $\inf \{ \, U(f, P) \, \} = \sup \{ \, L(f, P) \, \}$ where $P$ is any ...
1
vote
1answer
44 views

Intuition. Cauchy criteron for Riemann integrability (Spivak pp 239, S. Abbott pp 189 thm 7.2.8)

1. Why $\inf U(f,P') \le U(f, P)$ and $\sup L(f, P') \ge L(f,P) $? I tried to research but I can't find where Spivak defined it $P'$? 2. Why are there two partitions P', P''? Not the same? ...
1
vote
1answer
107 views

Easiest proof $\sup A + \sup B ≤ \sup(A + B).$ No epsilons, sequences. (S.A. pp 18 q1.3.9d)

(question 2. http://webcache.googleusercontent.com/search?q=cache:DohoRC3-bU8J:www.maths.usyd.edu.au/u/UG/IM/MATH2962/r/PDF/tut01s.pdf) Essay By definition of A + B and sup(A + B), for all a ∈ A and ...
3
votes
1answer
68 views

If sup A < sup B, there exists an element b ∈ B that's an upper bound for A. (S.A. pp 18 q1.3.8)

My Figure: By definition of $\sup B$, $\sup B$ is an upper bound for $B$. Set $e = \sup B − \sup A > 0$. By Lemma 1.3.7, there exists an element $b ∈ B$ satisfying $\begin{align} & \sup B − ...
1
vote
0answers
59 views

Prove limit of modulus of quotient of two functions is infinity (S.A. pp 144 question 5.3.9)

We need to prove for all $M > 0$, there exists d such that $0 < |x−c| < d \implies |\frac{f(x) }{ g(x) }| ≥ M.$ Choose $d_1$ so that $0 < |x−c| < d_1 \implies \color{brown}{|f(x) − ...
1
vote
2answers
50 views

If $g'(c) \neq 0$, show $g(x) \neq g(c)$ for all $x \in V_d(c)$ (S.A. pp 144 q5.3.8)

Let $g : (a, b) \to R$ be differentiable at a point $c \in (a, b)$. $V_d(c) := \{ x \in R : |x - c| < d \}$. First Case $g'(c) > 0$: We cannot use the mean value theorem since we only know ...
1
vote
1answer
52 views

Length of a curve by integration: why won't flat segments do?

Maybe my question is a duplicate, but I guess I don't know the right terminology to find it elsewhere. I would be happy to delete it if someone can point out a duplicate. From elementary calculus, ...
1
vote
1answer
92 views

Cauchy's Generalized Mean Value Theorem. Required function. (S.A. pp 140 t5.3.5)

Cohen, Henle. Calculus pp 827, (http://www.vias.org/calculus/09_infinite_series_10_06.html) I revised the footnote in pp 14 http://www.math.uga.edu/~pete/2400calc2.pdf. This theorem can be ...
2
votes
0answers
87 views

Intuition. If f differentiable and $\lim_{x\to0} f'(x) = L$, then $f'(0) = L$. (S.A. pp 137 q5.2.8c,d) [duplicate]

True/False. (c) If $f$ is differentiable on an interval containing zero and if $\lim_{x\to0} f(x) = L$, then $f(0) = L$. (d) Drop the assumption that $f'(0)$ necessarily exists. Not a duplicate of ...
1
vote
1answer
180 views

If f' exists and f'(c) > 0 then f'(x) > 0 for all |x - c| < d for some d. (S.A. pp 137 question 5.2.8b)

If $f'$ exists on an open interval, and there is some point $c$ where $f'(c) > 0$, then there exists a d-neighborhood $\{x \in \mathbb{R} : |x - c| < d\} = V_d(c)$ around c in which $f'(x) > ...
2
votes
0answers
65 views

Function on $\mathbb{R}$ differentiable at a single point (S.A. pp 136 question 5.2.3)

By imitating the Dirichlet constructions in Section 4.1, construct a function on R that is differentiable at a single point. Tried and assayed. 1. How can you calculate this construction? Where ...
2
votes
0answers
47 views

Why always $\delta = 1/n$? Negation of Continuity and Uniform Continuity? (S.A. pp 117 T4.4.6)

These proofs about negation of continuity and uniform continuity proofs always invoke $d = 1/n$. Where did this emanate from? I know $\lim _{n\rightarrow \infty }\frac {1} {n}=0$. Why not something ...
3
votes
0answers
33 views

Intution, Figure. Negation of Continuity and Uniform Continuity (S.A. pp 117 T4.4.6)

Every time I need negation, I have to write out all the logical symbols to negate manually. I know how to determine these negations myself. But I want to compehend intuition or figure like ...
5
votes
1answer
181 views

Equivalences of continuity, sequential convergence iff limit (S.A. pp 106 t4.2.3, 110 t4.3.2)

1. This post became too long, ergo I moved this here. 2. I questioned anew here. How does $\color{red}{(I) \implies (III)}$? This contradicts $a \le b \not \implies \Leftarrow a < b$. 3. ...
4
votes
1answer
138 views

Prove nth root of k exists with supremum. (Abbott pp 27 1.4.6b) [closed]

(Ulrich Daepp. Reading, Writing, and Proving. edition 2. pp 133 Theorem 13.2) Modus Operandi. The basic idea is that the nth root = supremum of $A = \left\{ w\in \mathbb{R} ^{+}:w^{n} < ...
3
votes
4answers
282 views

Intuition: If $a\leq b+\epsilon$ for all $\epsilon>0$ then $a\leq b$?

I am reading Tom Apostol's Analysis and come across this theorem. Should $a \leq b$ if $a\leq b+\epsilon$ for all $\epsilon >0$? I don't doubt the proof in the book but I don't understand the ...
3
votes
0answers
63 views

Intuition on Axiom of Completeness

♪ (J. Stewart. Calculus 6th ed. pp 682) Axiom of Completeness = AoC = A nonempty set of real numbers that has an upper bound has a least upper bound. AoC is an expression of the fact that there ...
2
votes
1answer
216 views

Direct proof. Square root function uniformly continuous on $[0, \infty)$ (S.A. pp 119 4.4.8)

(http://math.stanford.edu/~ksound/Math171S10/Hw8Sol_171.pdf) Prove for all $e > 0,$ there exists $d > 0$ : for all $x, y \ge 0$, $|x - y| < d \implies |\sqrt{x} - \sqrt{y}| < e$. (a) ...
5
votes
2answers
91 views

Intuition for differentiating beneath the integral

I apologize in advance for a vague question. There is a theorem: If both $f(x,s)$ and $\partial _sf(x,s)$ are continuous in $x$ and $s$, then $$\partial_s\int_a^bf(x,s)\,dx=\int_a^b ...
11
votes
1answer
232 views

“Why” is $[\mathbb{C}:\mathbb{R}] < \infty$?

Obviously this question is a little open-ended. A lot of complex analysis seems to work primarily because we can view $\mathbb{C}$ as a finite-dimensional $\mathbb{R}$-algebra, and apply analytic and ...
1
vote
1answer
38 views

What's behind the function $g(x)=\operatorname{inf}\{f(p)+d(x,p):p\in X\}$?

In several books on measure theory, I have seen the following problem: Suppose $(X,d)$ is a metric space, on which $f$ is a nonnegative lower semicontinuous function. Show that $f$ is the ...
2
votes
1answer
68 views

Finding $g_i:\mathbb{R}^n\to\mathbb R$ s.t $f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$

Let $f:\mathbb R^n\to\mathbb R$ differntiable and $f(0)=0$. Prove exist $g_i$ s.t for $x=(x_1,\dots,x_n):f(x)=\sum\limits_{i=1}^nx_i\cdot g_i(x)$. hint:$f(x)=\int\limits_0^1f\prime(tx)dt$. I dont ...
3
votes
0answers
79 views

A question on mathematical writing.

One of the problems I am grading this week is as follows: Given a simply connected bounded domain $\Omega$ on $\mathbb{R}^{2}$, prove that there exist a line that separates it into two parts of equal ...
0
votes
2answers
85 views

Lebesgue Measure Definition

Given a subset $A \subset \mathbb{R}$ with the length of an open interval $\mu_L(I_k) = b_k -a_k : I \doteq [a_k,b_k]$ The lebesgue measure is defined as $$ \lambda^{\ast} (A) \doteq \inf \Big\{ ...
2
votes
1answer
71 views

Equality and order in sets

Just started Baby Rudin and got struck in this. While defining order in sets, $<$ was introduced as a relation and for a set to be ordered the condition was: for all $x,y$ belonging to an ordered ...
4
votes
2answers
158 views

How does a geometry-oriented mind learn analysis?

I find it very difficult to understand analysis, because I can't find a way to learn it geometrically. To make my point clearer, let me take calculus as the example in contrast. I find calculus very ...
1
vote
1answer
58 views

$|f(x)|<|g(x)|$ and $\int g(x)<\infty\Rightarrow\ \int f(x)<\infty$

Let f,g continious functions on $[0,\infty)$ s.t $\forall x\in[0,\infty), |f(x)|\le|g(x)|$. Prove or give counterexample thtat if $\int_ 0^\infty g(x)dx<\infty$ then $\int_0^\infty ...
1
vote
2answers
283 views

A basic question in the definition of limit point

For any subset of $R$ with the usual distance metric, any point inside it is a limit point. Only when the set is discrete there may be a point inside it which is not a limit point. Is this correct ? ...