All Questions
Tagged with functions solution-verification
1,052
questions
4
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3
answers
591
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A strange confusion over a problem of continuity in Multivariate Calculus.
For $\beta\in\Bbb R,$ define $$f(x,y)= \begin{cases}\cfrac{x^2|x|^{\beta}y}{x^4+y^2},&x\neq 0 \\ 0, &x=0 \end{cases}$$
Prove that at $(0,0)$ the function is discontinuous if $\beta=0.$
My ...
- 3,040
0
votes
0
answers
42
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Another request for feedback on correctness and style (Velleman. How to Prove it. Exercise 5.4.2.d)
Level: first-year undergraduate learning proof writing.
Questions:
Is my proof of the amended claim correct?
How is the style of my proof? Have I provided enough scaffolding for the assumed level? Or ...
1
vote
1
answer
45
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Asking for feedback on correctness and style (Velleman. How to Prove it. Exercise 5.4.1.b)
Level: first-year undergraduate learning proof writing.
Questions:
Is my proof of the amended claim correct?
How is the style? Have I provided enough scaffolding for the assumed level? Or is it too ...
7
votes
4
answers
1k
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Disproving surjectivity of $f : \Bbb Z \times \Bbb Z \rightarrow \Bbb Z$, $f(u,v) = 3u + 6v$
A function $f : \Bbb Z \times \Bbb Z \rightarrow \Bbb Z$ is defined as $f(u,v) = 3u + 6v.$
Is the function surjective? Prove it.
I had the following proof.
Proof
Pick $x = 2$, then $3u + 6v = 2 \...
- 71
0
votes
0
answers
74
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Velleman 5.1.18. Set theory exercise - is my proof correct?
[edit: reformatting for ease of reading in the hopes of getting a response on whether my latest revision is correct]
Context: I'm a first-year undergraduate working my way through How to Prove It? by ...
0
votes
1
answer
68
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unsure of my solution to Tao Analysis I 4th ed 3.5.2 (axiomatic set construction, cartesian products, power sets)
I am a self-teaching beginner and unfamiliar with proofs - in particular I find proposals that are intuitively true harder to prove as my brain assumes too much or skips steps.
I'd like help with my ...
- 3,325
0
votes
1
answer
54
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Demonstrate that A is a countable set.
Question: Let $A = \{x \in \Bbb N \mid \exists y(x = 2y \lor x = y^2)\}$. Construct a surjective mapping $ f: \mathbb{N} \rightarrow A $. By doing this, you demonstrate that A is a countable set.
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- 111
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votes
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answers
41
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Show that the interval ⟨2, 5⟩ ⊆ ℝ⁺ is an uncountable set.
Question: Show that the interval $⟨2, 5⟩ ⊆ ℝ⁺$ is an uncountable set.\
To show that the interval $ \langle 2, 5 \rangle \subseteq \mathbb{R}^+ $ is an uncountable set, we can use Cantor's diagonal ...
- 111
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0
answers
68
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$f(x)=\lim_{n\to\infty}\left(\frac{n^n}{n!}\prod_{r=1}^n\frac{x^2+\frac{n^2}{r^2}}{x^3+\frac{n^3}{r^3}}\right)^{\frac xn}$
$$f(x)=\lim_{n\to\infty}\left(\frac{n^n}{n!}\prod_{r=1}^n\frac{x^2+\frac{n^2}{r^2}}{x^3+\frac{n^3}{r^3}}\right)^{\frac xn}$$What is the monotonicity of $f(x),x\gt0$
I tried in the following way [...
- 851
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votes
1
answer
11
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A clarification regarding the definition of uniform continuity of a function defined in a subset of $\mathbb R.$
Let $f:A\to \Bbb R$ where $A\subseteq \Bbb R$. We say that, $f$ is uniformly continuous on $A$ if for any $\epsilon\gt 0$ there exists $\delta(\epsilon)=\delta\gt 0$ such that for any $x_1,x_2\in A$ ...
- 3,040
-2
votes
1
answer
59
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Prove using $\epsilon$-$\delta$ definition that if $g$ continous in $a$ and that if $f$ continuous in $g(a)$ then $f\circ g$ is continuous in $a$ [duplicate]
Question:
Prove using $\epsilon$-$\delta$ definition that if $g$ continous in $a$ and that if $f$ continuous in $g(a)$ then $f\circ g$ is continuous in $a$
Rem: I know that there is a similar question ...
- 719
0
votes
1
answer
61
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Computing $h(h(x))$ where $h (x) = \lfloor 5x - 2 \rfloor$
In Velleman's "Calculus: a Rigorous Course," Example 9 from Section 1.3 tasks us with computing $ h(h(x)) $, where $ h(x) = \lfloor 5x - 2 \rfloor $.
My initial solution:
\begin{align*}
h(\...
- 2,357
-1
votes
1
answer
114
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Let f be a function f: A → B, and let A1, A2 ⊆ A. Then f(A1 ∩ A2) = f(A1) ∩ f(A2).
Question:
Let f be a function f: A → B, and let A1, A2 ⊆ A. Then f(A1 ∩ A2) = f(A1) ∩ f(A2).
Answer:
$$ f(A_1 \cap A_2) \subseteq f(A_1) \cap f(A_2) :$$ Let $ y $ be an arbitrary element in $ f(A_1 \...
- 111
-3
votes
3
answers
56
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Prove that the sum of a convex function and a concave function is convex.
I was trying to prove, for the sake of curiosity, if the sum of a convex and a concave functions is convex, so i tried to do the following:
Let f: R -> R and g: R -> R.
f is convex and g is ...
3
votes
6
answers
353
views
An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?
An attempt for approximating the logarithm function $\ln(x)$: Could be extended for big numbers?
PS: Thanks everyone for your comments and interesting answers showing how currently the logarithm ...
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